Find the Upper and Lower Bounds x^4-7x^3+3x^2+63x-108

$x^{4} - 7 x^{3} + 3 x^{2} + 63 x - 108$

Step 1

Find every combination of $\pm \frac{p}{q}$ .

If a polynomial function has integer coefficients, then every rational zero will have the form $\frac{p}{q}$ where $p$ is a factor of the constant and $q$ is a factor of the leading coefficient.

$p = \pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 9, \pm 12, \pm 18, \pm 27, \pm 36, \pm 54, \pm 108$

$q = \pm 1$

Find every combination of $\pm \frac{p}{q}$ . These are the possible roots of the polynomial function.

$\pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 9, \pm 12, \pm 18, \pm 27, \pm 36, \pm 54, \pm 108$

Step 2

Apply synthetic division on $\frac{x^{4} - 7 x^{3} + 3 x^{2} + 63 x - 108}{x + 3}$ when $x = - 3$ .

Place the numbers representing the divisor and the dividend into a division-like configuration.

$- 3$	$1$	$- 7$	$3$	$63$	$- 108$
	\xa0				\xa0

The first number in the dividend $(1)$ is put into the first position of the result area (below the horizontal line).

$- 3$	$1$	$- 7$	$3$	$63$	$- 108$
	\xa0				\xa0
	$1$

Multiply the newest entry in the result $(1)$ by the divisor $(- 3)$ and place the result of $(- 3)$ under the next term in the dividend $(- 7)$ .

$- 3$	$1$	$- 7$	$3$	$63$	$- 108$
	\xa0	$- 3$			\xa0
	$1$

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$- 3$	$1$	$- 7$	$3$	$63$	$- 108$
	\xa0	$- 3$			\xa0
	$1$	$- 10$

Multiply the newest entry in the result $(- 10)$ by the divisor $(- 3)$ and place the result of $(30)$ under the next term in the dividend $(3)$ .

$- 3$	$1$	$- 7$	$3$	$63$	$- 108$
	\xa0	$- 3$	$30$		\xa0
	$1$	$- 10$

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$- 3$	$1$	$- 7$	$3$	$63$	$- 108$
	\xa0	$- 3$	$30$		\xa0
	$1$	$- 10$	$33$

Multiply the newest entry in the result $(33)$ by the divisor $(- 3)$ and place the result of $(- 99)$ under the next term in the dividend $(63)$ .

$- 3$	$1$	$- 7$	$3$	$63$	$- 108$
	\xa0	$- 3$	$30$	$- 99$	\xa0
	$1$	$- 10$	$33$

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$- 3$	$1$	$- 7$	$3$	$63$	$- 108$
	\xa0	$- 3$	$30$	$- 99$	\xa0
	$1$	$- 10$	$33$	$- 36$

Multiply the newest entry in the result $(- 36)$ by the divisor $(- 3)$ and place the result of $(108)$ under the next term in the dividend $(- 108)$ .

$- 3$	$1$	$- 7$	$3$	$63$	$- 108$
	\xa0	$- 3$	$30$	$- 99$	$108$
	$1$	$- 10$	$33$	$- 36$

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$- 3$	$1$	$- 7$	$3$	$63$	$- 108$
	\xa0	$- 3$	$30$	$- 99$	$108$
	$1$	$- 10$	$33$	$- 36$	$0$

All numbers except the last become the coefficients of the quotient polynomial. The last value in the result line is the remainder.

$1 x^{3} + - 10 x^{2} + (33) x - 36$

Simplify the quotient polynomial.

$x^{3} - 10 x^{2} + 33 x - 36$

Step 3

Since $- 3 < 0$ and the signs in the bottom row of the synthetic division alternate sign, $- 3$ is a lower bound for the real roots of the function.

Lower Bound: $- 3$

Step 4

Apply synthetic division on $\frac{x^{4} - 7 x^{3} + 3 x^{2} + 63 x - 108}{x + 4}$ when $x = - 4$ .

Place the numbers representing the divisor and the dividend into a division-like configuration.

$- 4$	$1$	$- 7$	$3$	$63$	$- 108$
	\xa0				\xa0

The first number in the dividend $(1)$ is put into the first position of the result area (below the horizontal line).

$- 4$	$1$	$- 7$	$3$	$63$	$- 108$
	\xa0				\xa0
	$1$

Multiply the newest entry in the result $(1)$ by the divisor $(- 4)$ and place the result of $(- 4)$ under the next term in the dividend $(- 7)$ .

$- 4$	$1$	$- 7$	$3$	$63$	$- 108$
	\xa0	$- 4$			\xa0
	$1$

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$- 4$	$1$	$- 7$	$3$	$63$	$- 108$
	\xa0	$- 4$			\xa0
	$1$	$- 11$

Multiply the newest entry in the result $(- 11)$ by the divisor $(- 4)$ and place the result of $(44)$ under the next term in the dividend $(3)$ .

$- 4$	$1$	$- 7$	$3$	$63$	$- 108$
	\xa0	$- 4$	$44$		\xa0
	$1$	$- 11$

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$- 4$	$1$	$- 7$	$3$	$63$	$- 108$
	\xa0	$- 4$	$44$		\xa0
	$1$	$- 11$	$47$

Multiply the newest entry in the result $(47)$ by the divisor $(- 4)$ and place the result of $(- 188)$ under the next term in the dividend $(63)$ .

$- 4$	$1$	$- 7$	$3$	$63$	$- 108$
	\xa0	$- 4$	$44$	$- 188$	\xa0
	$1$	$- 11$	$47$

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$- 4$	$1$	$- 7$	$3$	$63$	$- 108$
	\xa0	$- 4$	$44$	$- 188$	\xa0
	$1$	$- 11$	$47$	$- 125$

Multiply the newest entry in the result $(- 125)$ by the divisor $(- 4)$ and place the result of $(500)$ under the next term in the dividend $(- 108)$ .

$- 4$	$1$	$- 7$	$3$	$63$	$- 108$
	\xa0	$- 4$	$44$	$- 188$	$500$
	$1$	$- 11$	$47$	$- 125$

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$- 4$	$1$	$- 7$	$3$	$63$	$- 108$
	\xa0	$- 4$	$44$	$- 188$	$500$
	$1$	$- 11$	$47$	$- 125$	$392$

All numbers except the last become the coefficients of the quotient polynomial. The last value in the result line is the remainder.

$1 x^{3} + - 11 x^{2} + (47) x - 125 + \frac{392}{x + 4}$

Simplify the quotient polynomial.

$x^{3} - 11 x^{2} + 47 x - 125 + \frac{392}{x + 4}$

Step 5

Since $- 4 < 0$ and the signs in the bottom row of the synthetic division alternate sign, $- 4$ is a lower bound for the real roots of the function.

Lower Bound: $- 4$

Step 6

Apply synthetic division on $\frac{x^{4} - 7 x^{3} + 3 x^{2} + 63 x - 108}{x + 6}$ when $x = - 6$ .

Place the numbers representing the divisor and the dividend into a division-like configuration.

$- 6$	$1$	$- 7$	$3$	$63$	$- 108$
	\xa0				\xa0

The first number in the dividend $(1)$ is put into the first position of the result area (below the horizontal line).

$- 6$	$1$	$- 7$	$3$	$63$	$- 108$
	\xa0				\xa0
	$1$

Multiply the newest entry in the result $(1)$ by the divisor $(- 6)$ and place the result of $(- 6)$ under the next term in the dividend $(- 7)$ .

$- 6$	$1$	$- 7$	$3$	$63$	$- 108$
	\xa0	$- 6$			\xa0
	$1$

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$- 6$	$1$	$- 7$	$3$	$63$	$- 108$
	\xa0	$- 6$			\xa0
	$1$	$- 13$

Multiply the newest entry in the result $(- 13)$ by the divisor $(- 6)$ and place the result of $(78)$ under the next term in the dividend $(3)$ .

$- 6$	$1$	$- 7$	$3$	$63$	$- 108$
	\xa0	$- 6$	$78$		\xa0
	$1$	$- 13$

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$- 6$	$1$	$- 7$	$3$	$63$	$- 108$
	\xa0	$- 6$	$78$		\xa0
	$1$	$- 13$	$81$

Multiply the newest entry in the result $(81)$ by the divisor $(- 6)$ and place the result of $(- 486)$ under the next term in the dividend $(63)$ .

$- 6$	$1$	$- 7$	$3$	$63$	$- 108$
	\xa0	$- 6$	$78$	$- 486$	\xa0
	$1$	$- 13$	$81$

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$- 6$	$1$	$- 7$	$3$	$63$	$- 108$
	\xa0	$- 6$	$78$	$- 486$	\xa0
	$1$	$- 13$	$81$	$- 423$

Multiply the newest entry in the result $(- 423)$ by the divisor $(- 6)$ and place the result of $(2538)$ under the next term in the dividend $(- 108)$ .

$- 6$	$1$	$- 7$	$3$	$63$	$- 108$
	\xa0	$- 6$	$78$	$- 486$	$2538$
	$1$	$- 13$	$81$	$- 423$

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$- 6$	$1$	$- 7$	$3$	$63$	$- 108$
	\xa0	$- 6$	$78$	$- 486$	$2538$
	$1$	$- 13$	$81$	$- 423$	$2430$

All numbers except the last become the coefficients of the quotient polynomial. The last value in the result line is the remainder.

$1 x^{3} + - 13 x^{2} + (81) x - 423 + \frac{2430}{x + 6}$

Simplify the quotient polynomial.

$x^{3} - 13 x^{2} + 81 x - 423 + \frac{2430}{x + 6}$

Step 7

Since $- 6 < 0$ and the signs in the bottom row of the synthetic division alternate sign, $- 6$ is a lower bound for the real roots of the function.

Lower Bound: $- 6$

Step 8

Apply synthetic division on $\frac{x^{4} - 7 x^{3} + 3 x^{2} + 63 x - 108}{x - 9}$ when $x = 9$ .

Place the numbers representing the divisor and the dividend into a division-like configuration.

$9$	$1$	$- 7$	$3$	$63$	$- 108$
	\xa0				\xa0

The first number in the dividend $(1)$ is put into the first position of the result area (below the horizontal line).

$9$	$1$	$- 7$	$3$	$63$	$- 108$
	\xa0				\xa0
	$1$

Multiply the newest entry in the result $(1)$ by the divisor $(9)$ and place the result of $(9)$ under the next term in the dividend $(- 7)$ .

$9$	$1$	$- 7$	$3$	$63$	$- 108$
	\xa0	$9$			\xa0
	$1$

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$9$	$1$	$- 7$	$3$	$63$	$- 108$
	\xa0	$9$			\xa0
	$1$	$2$

Multiply the newest entry in the result $(2)$ by the divisor $(9)$ and place the result of $(18)$ under the next term in the dividend $(3)$ .

$9$	$1$	$- 7$	$3$	$63$	$- 108$
	\xa0	$9$	$18$		\xa0
	$1$	$2$

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$9$	$1$	$- 7$	$3$	$63$	$- 108$
	\xa0	$9$	$18$		\xa0
	$1$	$2$	$21$

Multiply the newest entry in the result $(21)$ by the divisor $(9)$ and place the result of $(189)$ under the next term in the dividend $(63)$ .

$9$	$1$	$- 7$	$3$	$63$	$- 108$
	\xa0	$9$	$18$	$189$	\xa0
	$1$	$2$	$21$

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$9$	$1$	$- 7$	$3$	$63$	$- 108$
	\xa0	$9$	$18$	$189$	\xa0
	$1$	$2$	$21$	$252$

Multiply the newest entry in the result $(252)$ by the divisor $(9)$ and place the result of $(2268)$ under the next term in the dividend $(- 108)$ .

$9$	$1$	$- 7$	$3$	$63$	$- 108$
	\xa0	$9$	$18$	$189$	$2268$
	$1$	$2$	$21$	$252$

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$9$	$1$	$- 7$	$3$	$63$	$- 108$
	\xa0	$9$	$18$	$189$	$2268$
	$1$	$2$	$21$	$252$	$2160$

All numbers except the last become the coefficients of the quotient polynomial. The last value in the result line is the remainder.

$1 x^{3} + 2 x^{2} + (21) x + 252 + \frac{2160}{x - 9}$

Simplify the quotient polynomial.

$x^{3} + 2 x^{2} + 21 x + 252 + \frac{2160}{x - 9}$

Step 9

Since $9 > 0$ and all of the signs in the bottom row of the synthetic division are positive, $9$ is an upper bound for the real roots of the function.

Upper Bound: $9$

Step 10

Apply synthetic division on $\frac{x^{4} - 7 x^{3} + 3 x^{2} + 63 x - 108}{x + 9}$ when $x = - 9$ .

Place the numbers representing the divisor and the dividend into a division-like configuration.

$- 9$	$1$	$- 7$	$3$	$63$	$- 108$
	\xa0				\xa0

The first number in the dividend $(1)$ is put into the first position of the result area (below the horizontal line).

$- 9$	$1$	$- 7$	$3$	$63$	$- 108$
	\xa0				\xa0
	$1$

Multiply the newest entry in the result $(1)$ by the divisor $(- 9)$ and place the result of $(- 9)$ under the next term in the dividend $(- 7)$ .

$- 9$	$1$	$- 7$	$3$	$63$	$- 108$
	\xa0	$- 9$			\xa0
	$1$

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$- 9$	$1$	$- 7$	$3$	$63$	$- 108$
	\xa0	$- 9$			\xa0
	$1$	$- 16$

Multiply the newest entry in the result $(- 16)$ by the divisor $(- 9)$ and place the result of $(144)$ under the next term in the dividend $(3)$ .

$- 9$	$1$	$- 7$	$3$	$63$	$- 108$
	\xa0	$- 9$	$144$		\xa0
	$1$	$- 16$

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$- 9$	$1$	$- 7$	$3$	$63$	$- 108$
	\xa0	$- 9$	$144$		\xa0
	$1$	$- 16$	$147$

Multiply the newest entry in the result $(147)$ by the divisor $(- 9)$ and place the result of $(- 1323)$ under the next term in the dividend $(63)$ .

$- 9$	$1$	$- 7$	$3$	$63$	$- 108$
	\xa0	$- 9$	$144$	$- 1323$	\xa0
	$1$	$- 16$	$147$

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$- 9$	$1$	$- 7$	$3$	$63$	$- 108$
	\xa0	$- 9$	$144$	$- 1323$	\xa0
	$1$	$- 16$	$147$	$- 1260$

Multiply the newest entry in the result $(- 1260)$ by the divisor $(- 9)$ and place the result of $(11340)$ under the next term in the dividend $(- 108)$ .

$- 9$	$1$	$- 7$	$3$	$63$	$- 108$
	\xa0	$- 9$	$144$	$- 1323$	$11340$
	$1$	$- 16$	$147$	$- 1260$

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$- 9$	$1$	$- 7$	$3$	$63$	$- 108$
	\xa0	$- 9$	$144$	$- 1323$	$11340$
	$1$	$- 16$	$147$	$- 1260$	$11232$

All numbers except the last become the coefficients of the quotient polynomial. The last value in the result line is the remainder.

$1 x^{3} + - 16 x^{2} + (147) x - 1260 + \frac{11232}{x + 9}$

Simplify the quotient polynomial.

$x^{3} - 16 x^{2} + 147 x - 1260 + \frac{11232}{x + 9}$

Step 11

Since $- 9 < 0$ and the signs in the bottom row of the synthetic division alternate sign, $- 9$ is a lower bound for the real roots of the function.

Lower Bound: $- 9$

Step 12

Apply synthetic division on $\frac{x^{4} - 7 x^{3} + 3 x^{2} + 63 x - 108}{x - 12}$ when $x = 12$ .

Place the numbers representing the divisor and the dividend into a division-like configuration.

$12$	$1$	$- 7$	$3$	$63$	$- 108$
	\xa0				\xa0

The first number in the dividend $(1)$ is put into the first position of the result area (below the horizontal line).

$12$	$1$	$- 7$	$3$	$63$	$- 108$
	\xa0				\xa0
	$1$

Multiply the newest entry in the result $(1)$ by the divisor $(12)$ and place the result of $(12)$ under the next term in the dividend $(- 7)$ .

$12$	$1$	$- 7$	$3$	$63$	$- 108$
	\xa0	$12$			\xa0
	$1$

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$12$	$1$	$- 7$	$3$	$63$	$- 108$
	\xa0	$12$			\xa0
	$1$	$5$

Multiply the newest entry in the result $(5)$ by the divisor $(12)$ and place the result of $(60)$ under the next term in the dividend $(3)$ .

$12$	$1$	$- 7$	$3$	$63$	$- 108$
	\xa0	$12$	$60$		\xa0
	$1$	$5$

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$12$	$1$	$- 7$	$3$	$63$	$- 108$
	\xa0	$12$	$60$		\xa0
	$1$	$5$	$63$

Multiply the newest entry in the result $(63)$ by the divisor $(12)$ and place the result of $(756)$ under the next term in the dividend $(63)$ .

$12$	$1$	$- 7$	$3$	$63$	$- 108$
	\xa0	$12$	$60$	$756$	\xa0
	$1$	$5$	$63$

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$12$	$1$	$- 7$	$3$	$63$	$- 108$
	\xa0	$12$	$60$	$756$	\xa0
	$1$	$5$	$63$	$819$

Multiply the newest entry in the result $(819)$ by the divisor $(12)$ and place the result of $(9828)$ under the next term in the dividend $(- 108)$ .

$12$	$1$	$- 7$	$3$	$63$	$- 108$
	\xa0	$12$	$60$	$756$	$9828$
	$1$	$5$	$63$	$819$

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$12$	$1$	$- 7$	$3$	$63$	$- 108$
	\xa0	$12$	$60$	$756$	$9828$
	$1$	$5$	$63$	$819$	$9720$

All numbers except the last become the coefficients of the quotient polynomial. The last value in the result line is the remainder.

$1 x^{3} + 5 x^{2} + (63) x + 819 + \frac{9720}{x - 12}$

Simplify the quotient polynomial.

$x^{3} + 5 x^{2} + 63 x + 819 + \frac{9720}{x - 12}$

Step 13

Since $12 > 0$ and all of the signs in the bottom row of the synthetic division are positive, $12$ is an upper bound for the real roots of the function.

Upper Bound: $12$

Step 14

Apply synthetic division on $\frac{x^{4} - 7 x^{3} + 3 x^{2} + 63 x - 108}{x + 12}$ when $x = - 12$ .

Place the numbers representing the divisor and the dividend into a division-like configuration.

$- 12$	$1$	$- 7$	$3$	$63$	$- 108$
	\xa0				\xa0

The first number in the dividend $(1)$ is put into the first position of the result area (below the horizontal line).

$- 12$	$1$	$- 7$	$3$	$63$	$- 108$
	\xa0				\xa0
	$1$

Multiply the newest entry in the result $(1)$ by the divisor $(- 12)$ and place the result of $(- 12)$ under the next term in the dividend $(- 7)$ .

$- 12$	$1$	$- 7$	$3$	$63$	$- 108$
	\xa0	$- 12$			\xa0
	$1$

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$- 12$	$1$	$- 7$	$3$	$63$	$- 108$
	\xa0	$- 12$			\xa0
	$1$	$- 19$

Multiply the newest entry in the result $(- 19)$ by the divisor $(- 12)$ and place the result of $(228)$ under the next term in the dividend $(3)$ .

$- 12$	$1$	$- 7$	$3$	$63$	$- 108$
	\xa0	$- 12$	$228$		\xa0
	$1$	$- 19$

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$- 12$	$1$	$- 7$	$3$	$63$	$- 108$
	\xa0	$- 12$	$228$		\xa0
	$1$	$- 19$	$231$

Multiply the newest entry in the result $(231)$ by the divisor $(- 12)$ and place the result of $(- 2772)$ under the next term in the dividend $(63)$ .

$- 12$	$1$	$- 7$	$3$	$63$	$- 108$
	\xa0	$- 12$	$228$	$- 2772$	\xa0
	$1$	$- 19$	$231$

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$- 12$	$1$	$- 7$	$3$	$63$	$- 108$
	\xa0	$- 12$	$228$	$- 2772$	\xa0
	$1$	$- 19$	$231$	$- 2709$

Multiply the newest entry in the result $(- 2709)$ by the divisor $(- 12)$ and place the result of $(32508)$ under the next term in the dividend $(- 108)$ .

$- 12$	$1$	$- 7$	$3$	$63$	$- 108$
	\xa0	$- 12$	$228$	$- 2772$	$32508$
	$1$	$- 19$	$231$	$- 2709$

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$- 12$	$1$	$- 7$	$3$	$63$	$- 108$
	\xa0	$- 12$	$228$	$- 2772$	$32508$
	$1$	$- 19$	$231$	$- 2709$	$32400$

All numbers except the last become the coefficients of the quotient polynomial. The last value in the result line is the remainder.

$1 x^{3} + - 19 x^{2} + (231) x - 2709 + \frac{32400}{x + 12}$

Simplify the quotient polynomial.

$x^{3} - 19 x^{2} + 231 x - 2709 + \frac{32400}{x + 12}$

Step 15

Since $- 12 < 0$ and the signs in the bottom row of the synthetic division alternate sign, $- 12$ is a lower bound for the real roots of the function.

Lower Bound: $- 12$

Step 16

Apply synthetic division on $\frac{x^{4} - 7 x^{3} + 3 x^{2} + 63 x - 108}{x - 18}$ when $x = 18$ .

Place the numbers representing the divisor and the dividend into a division-like configuration.

$18$	$1$	$- 7$	$3$	$63$	$- 108$
	\xa0				\xa0

The first number in the dividend $(1)$ is put into the first position of the result area (below the horizontal line).

$18$	$1$	$- 7$	$3$	$63$	$- 108$
	\xa0				\xa0
	$1$

Multiply the newest entry in the result $(1)$ by the divisor $(18)$ and place the result of $(18)$ under the next term in the dividend $(- 7)$ .

$18$	$1$	$- 7$	$3$	$63$	$- 108$
	\xa0	$18$			\xa0
	$1$

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$18$	$1$	$- 7$	$3$	$63$	$- 108$
	\xa0	$18$			\xa0
	$1$	$11$

Multiply the newest entry in the result $(11)$ by the divisor $(18)$ and place the result of $(198)$ under the next term in the dividend $(3)$ .

$18$	$1$	$- 7$	$3$	$63$	$- 108$
	\xa0	$18$	$198$		\xa0
	$1$	$11$

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$18$	$1$	$- 7$	$3$	$63$	$- 108$
	\xa0	$18$	$198$		\xa0
	$1$	$11$	$201$

Multiply the newest entry in the result $(201)$ by the divisor $(18)$ and place the result of $(3618)$ under the next term in the dividend $(63)$ .

$18$	$1$	$- 7$	$3$	$63$	$- 108$
	\xa0	$18$	$198$	$3618$	\xa0
	$1$	$11$	$201$

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$18$	$1$	$- 7$	$3$	$63$	$- 108$
	\xa0	$18$	$198$	$3618$	\xa0
	$1$	$11$	$201$	$3681$

Multiply the newest entry in the result $(3681)$ by the divisor $(18)$ and place the result of $(66258)$ under the next term in the dividend $(- 108)$ .

$18$	$1$	$- 7$	$3$	$63$	$- 108$
	\xa0	$18$	$198$	$3618$	$66258$
	$1$	$11$	$201$	$3681$

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$18$	$1$	$- 7$	$3$	$63$	$- 108$
	\xa0	$18$	$198$	$3618$	$66258$
	$1$	$11$	$201$	$3681$	$66150$

All numbers except the last become the coefficients of the quotient polynomial. The last value in the result line is the remainder.

$1 x^{3} + 11 x^{2} + (201) x + 3681 + \frac{66150}{x - 18}$

Simplify the quotient polynomial.

$x^{3} + 11 x^{2} + 201 x + 3681 + \frac{66150}{x - 18}$

Step 17

Since $18 > 0$ and all of the signs in the bottom row of the synthetic division are positive, $18$ is an upper bound for the real roots of the function.

Upper Bound: $18$

Step 18

Apply synthetic division on $\frac{x^{4} - 7 x^{3} + 3 x^{2} + 63 x - 108}{x + 18}$ when $x = - 18$ .

Place the numbers representing the divisor and the dividend into a division-like configuration.

$- 18$	$1$	$- 7$	$3$	$63$	$- 108$
	\xa0				\xa0

The first number in the dividend $(1)$ is put into the first position of the result area (below the horizontal line).

$- 18$	$1$	$- 7$	$3$	$63$	$- 108$
	\xa0				\xa0
	$1$

Multiply the newest entry in the result $(1)$ by the divisor $(- 18)$ and place the result of $(- 18)$ under the next term in the dividend $(- 7)$ .

$- 18$	$1$	$- 7$	$3$	$63$	$- 108$
	\xa0	$- 18$			\xa0
	$1$

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$- 18$	$1$	$- 7$	$3$	$63$	$- 108$
	\xa0	$- 18$			\xa0
	$1$	$- 25$

Multiply the newest entry in the result $(- 25)$ by the divisor $(- 18)$ and place the result of $(450)$ under the next term in the dividend $(3)$ .

$- 18$	$1$	$- 7$	$3$	$63$	$- 108$
	\xa0	$- 18$	$450$		\xa0
	$1$	$- 25$

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$- 18$	$1$	$- 7$	$3$	$63$	$- 108$
	\xa0	$- 18$	$450$		\xa0
	$1$	$- 25$	$453$

Multiply the newest entry in the result $(453)$ by the divisor $(- 18)$ and place the result of $(- 8154)$ under the next term in the dividend $(63)$ .

$- 18$	$1$	$- 7$	$3$	$63$	$- 108$
	\xa0	$- 18$	$450$	$- 8154$	\xa0
	$1$	$- 25$	$453$

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$- 18$	$1$	$- 7$	$3$	$63$	$- 108$
	\xa0	$- 18$	$450$	$- 8154$	\xa0
	$1$	$- 25$	$453$	$- 8091$

Multiply the newest entry in the result $(- 8091)$ by the divisor $(- 18)$ and place the result of $(145638)$ under the next term in the dividend $(- 108)$ .

$- 18$	$1$	$- 7$	$3$	$63$	$- 108$
	\xa0	$- 18$	$450$	$- 8154$	$145638$
	$1$	$- 25$	$453$	$- 8091$

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$- 18$	$1$	$- 7$	$3$	$63$	$- 108$
	\xa0	$- 18$	$450$	$- 8154$	$145638$
	$1$	$- 25$	$453$	$- 8091$	$145530$

All numbers except the last become the coefficients of the quotient polynomial. The last value in the result line is the remainder.

$1 x^{3} + - 25 x^{2} + (453) x - 8091 + \frac{145530}{x + 18}$

Simplify the quotient polynomial.

$x^{3} - 25 x^{2} + 453 x - 8091 + \frac{145530}{x + 18}$

Step 19

Since $- 18 < 0$ and the signs in the bottom row of the synthetic division alternate sign, $- 18$ is a lower bound for the real roots of the function.

Lower Bound: $- 18$

Step 20

Apply synthetic division on $\frac{x^{4} - 7 x^{3} + 3 x^{2} + 63 x - 108}{x - 27}$ when $x = 27$ .

Place the numbers representing the divisor and the dividend into a division-like configuration.

$27$	$1$	$- 7$	$3$	$63$	$- 108$
	\xa0				\xa0

The first number in the dividend $(1)$ is put into the first position of the result area (below the horizontal line).

$27$	$1$	$- 7$	$3$	$63$	$- 108$
	\xa0				\xa0
	$1$

Multiply the newest entry in the result $(1)$ by the divisor $(27)$ and place the result of $(27)$ under the next term in the dividend $(- 7)$ .

$27$	$1$	$- 7$	$3$	$63$	$- 108$
	\xa0	$27$			\xa0
	$1$

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$27$	$1$	$- 7$	$3$	$63$	$- 108$
	\xa0	$27$			\xa0
	$1$	$20$

Multiply the newest entry in the result $(20)$ by the divisor $(27)$ and place the result of $(540)$ under the next term in the dividend $(3)$ .

$27$	$1$	$- 7$	$3$	$63$	$- 108$
	\xa0	$27$	$540$		\xa0
	$1$	$20$

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$27$	$1$	$- 7$	$3$	$63$	$- 108$
	\xa0	$27$	$540$		\xa0
	$1$	$20$	$543$

Multiply the newest entry in the result $(543)$ by the divisor $(27)$ and place the result of $(14661)$ under the next term in the dividend $(63)$ .

$27$	$1$	$- 7$	$3$	$63$	$- 108$
	\xa0	$27$	$540$	$14661$	\xa0
	$1$	$20$	$543$

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$27$	$1$	$- 7$	$3$	$63$	$- 108$
	\xa0	$27$	$540$	$14661$	\xa0
	$1$	$20$	$543$	$14724$

Multiply the newest entry in the result $(14724)$ by the divisor $(27)$ and place the result of $(397548)$ under the next term in the dividend $(- 108)$ .

$27$	$1$	$- 7$	$3$	$63$	$- 108$
	\xa0	$27$	$540$	$14661$	$397548$
	$1$	$20$	$543$	$14724$

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$27$	$1$	$- 7$	$3$	$63$	$- 108$
	\xa0	$27$	$540$	$14661$	$397548$
	$1$	$20$	$543$	$14724$	$397440$

All numbers except the last become the coefficients of the quotient polynomial. The last value in the result line is the remainder.

$1 x^{3} + 20 x^{2} + (543) x + 14724 + \frac{397440}{x - 27}$

Simplify the quotient polynomial.

$x^{3} + 20 x^{2} + 543 x + 14724 + \frac{397440}{x - 27}$

Step 21

Since $27 > 0$ and all of the signs in the bottom row of the synthetic division are positive, $27$ is an upper bound for the real roots of the function.

Upper Bound: $27$

Step 22

Apply synthetic division on $\frac{x^{4} - 7 x^{3} + 3 x^{2} + 63 x - 108}{x + 27}$ when $x = - 27$ .

Place the numbers representing the divisor and the dividend into a division-like configuration.

$- 27$	$1$	$- 7$	$3$	$63$	$- 108$
	\xa0				\xa0

The first number in the dividend $(1)$ is put into the first position of the result area (below the horizontal line).

$- 27$	$1$	$- 7$	$3$	$63$	$- 108$
	\xa0				\xa0
	$1$

Multiply the newest entry in the result $(1)$ by the divisor $(- 27)$ and place the result of $(- 27)$ under the next term in the dividend $(- 7)$ .

$- 27$	$1$	$- 7$	$3$	$63$	$- 108$
	\xa0	$- 27$			\xa0
	$1$

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$- 27$	$1$	$- 7$	$3$	$63$	$- 108$
	\xa0	$- 27$			\xa0
	$1$	$- 34$

Multiply the newest entry in the result $(- 34)$ by the divisor $(- 27)$ and place the result of $(918)$ under the next term in the dividend $(3)$ .

$- 27$	$1$	$- 7$	$3$	$63$	$- 108$
	\xa0	$- 27$	$918$		\xa0
	$1$	$- 34$

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$- 27$	$1$	$- 7$	$3$	$63$	$- 108$
	\xa0	$- 27$	$918$		\xa0
	$1$	$- 34$	$921$

Multiply the newest entry in the result $(921)$ by the divisor $(- 27)$ and place the result of $(- 24867)$ under the next term in the dividend $(63)$ .

$- 27$	$1$	$- 7$	$3$	$63$	$- 108$
	\xa0	$- 27$	$918$	$- 24867$	\xa0
	$1$	$- 34$	$921$

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$- 27$	$1$	$- 7$	$3$	$63$	$- 108$
	\xa0	$- 27$	$918$	$- 24867$	\xa0
	$1$	$- 34$	$921$	$- 24804$

Multiply the newest entry in the result $(- 24804)$ by the divisor $(- 27)$ and place the result of $(669708)$ under the next term in the dividend $(- 108)$ .

$- 27$	$1$	$- 7$	$3$	$63$	$- 108$
	\xa0	$- 27$	$918$	$- 24867$	$669708$
	$1$	$- 34$	$921$	$- 24804$

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$- 27$	$1$	$- 7$	$3$	$63$	$- 108$
	\xa0	$- 27$	$918$	$- 24867$	$669708$
	$1$	$- 34$	$921$	$- 24804$	$669600$

All numbers except the last become the coefficients of the quotient polynomial. The last value in the result line is the remainder.

$1 x^{3} + - 34 x^{2} + (921) x - 24804 + \frac{669600}{x + 27}$

Simplify the quotient polynomial.

$x^{3} - 34 x^{2} + 921 x - 24804 + \frac{669600}{x + 27}$

Step 23

Since $- 27 < 0$ and the signs in the bottom row of the synthetic division alternate sign, $- 27$ is a lower bound for the real roots of the function.

Lower Bound: $- 27$

Step 24

Apply synthetic division on $\frac{x^{4} - 7 x^{3} + 3 x^{2} + 63 x - 108}{x - 36}$ when $x = 36$ .

Place the numbers representing the divisor and the dividend into a division-like configuration.

$36$	$1$	$- 7$	$3$	$63$	$- 108$
	\xa0				\xa0

The first number in the dividend $(1)$ is put into the first position of the result area (below the horizontal line).

$36$	$1$	$- 7$	$3$	$63$	$- 108$
	\xa0				\xa0
	$1$

Multiply the newest entry in the result $(1)$ by the divisor $(36)$ and place the result of $(36)$ under the next term in the dividend $(- 7)$ .

$36$	$1$	$- 7$	$3$	$63$	$- 108$
	\xa0	$36$			\xa0
	$1$

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$36$	$1$	$- 7$	$3$	$63$	$- 108$
	\xa0	$36$			\xa0
	$1$	$29$

Multiply the newest entry in the result $(29)$ by the divisor $(36)$ and place the result of $(1044)$ under the next term in the dividend $(3)$ .

$36$	$1$	$- 7$	$3$	$63$	$- 108$
	\xa0	$36$	$1044$		\xa0
	$1$	$29$

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$36$	$1$	$- 7$	$3$	$63$	$- 108$
	\xa0	$36$	$1044$		\xa0
	$1$	$29$	$1047$

Multiply the newest entry in the result $(1047)$ by the divisor $(36)$ and place the result of $(37692)$ under the next term in the dividend $(63)$ .

$36$	$1$	$- 7$	$3$	$63$	$- 108$
	\xa0	$36$	$1044$	$37692$	\xa0
	$1$	$29$	$1047$

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$36$	$1$	$- 7$	$3$	$63$	$- 108$
	\xa0	$36$	$1044$	$37692$	\xa0
	$1$	$29$	$1047$	$37755$

Multiply the newest entry in the result $(37755)$ by the divisor $(36)$ and place the result of $(1359180)$ under the next term in the dividend $(- 108)$ .

$36$	$1$	$- 7$	$3$	$63$	$- 108$
	\xa0	$36$	$1044$	$37692$	$1359180$
	$1$	$29$	$1047$	$37755$

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$36$	$1$	$- 7$	$3$	$63$	$- 108$
	\xa0	$36$	$1044$	$37692$	$1359180$
	$1$	$29$	$1047$	$37755$	$1359072$

All numbers except the last become the coefficients of the quotient polynomial. The last value in the result line is the remainder.

$1 x^{3} + 29 x^{2} + (1047) x + 37755 + \frac{1359072}{x - 36}$

Simplify the quotient polynomial.

$x^{3} + 29 x^{2} + 1047 x + 37755 + \frac{1359072}{x - 36}$

Step 25

Since $36 > 0$ and all of the signs in the bottom row of the synthetic division are positive, $36$ is an upper bound for the real roots of the function.

Upper Bound: $36$

Step 26

Apply synthetic division on $\frac{x^{4} - 7 x^{3} + 3 x^{2} + 63 x - 108}{x + 36}$ when $x = - 36$ .

Place the numbers representing the divisor and the dividend into a division-like configuration.

$- 36$	$1$	$- 7$	$3$	$63$	$- 108$
	\xa0				\xa0

The first number in the dividend $(1)$ is put into the first position of the result area (below the horizontal line).

$- 36$	$1$	$- 7$	$3$	$63$	$- 108$
	\xa0				\xa0
	$1$

Multiply the newest entry in the result $(1)$ by the divisor $(- 36)$ and place the result of $(- 36)$ under the next term in the dividend $(- 7)$ .

$- 36$	$1$	$- 7$	$3$	$63$	$- 108$
	\xa0	$- 36$			\xa0
	$1$

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$- 36$	$1$	$- 7$	$3$	$63$	$- 108$
	\xa0	$- 36$			\xa0
	$1$	$- 43$

Multiply the newest entry in the result $(- 43)$ by the divisor $(- 36)$ and place the result of $(1548)$ under the next term in the dividend $(3)$ .

$- 36$	$1$	$- 7$	$3$	$63$	$- 108$
	\xa0	$- 36$	$1548$		\xa0
	$1$	$- 43$

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$- 36$	$1$	$- 7$	$3$	$63$	$- 108$
	\xa0	$- 36$	$1548$		\xa0
	$1$	$- 43$	$1551$

Multiply the newest entry in the result $(1551)$ by the divisor $(- 36)$ and place the result of $(- 55836)$ under the next term in the dividend $(63)$ .

$- 36$	$1$	$- 7$	$3$	$63$	$- 108$
	\xa0	$- 36$	$1548$	$- 55836$	\xa0
	$1$	$- 43$	$1551$

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$- 36$	$1$	$- 7$	$3$	$63$	$- 108$
	\xa0	$- 36$	$1548$	$- 55836$	\xa0
	$1$	$- 43$	$1551$	$- 55773$

Multiply the newest entry in the result $(- 55773)$ by the divisor $(- 36)$ and place the result of $(2007828)$ under the next term in the dividend $(- 108)$ .

$- 36$	$1$	$- 7$	$3$	$63$	$- 108$
	\xa0	$- 36$	$1548$	$- 55836$	$2007828$
	$1$	$- 43$	$1551$	$- 55773$

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$- 36$	$1$	$- 7$	$3$	$63$	$- 108$
	\xa0	$- 36$	$1548$	$- 55836$	$2007828$
	$1$	$- 43$	$1551$	$- 55773$	$2007720$

All numbers except the last become the coefficients of the quotient polynomial. The last value in the result line is the remainder.

$1 x^{3} + - 43 x^{2} + (1551) x - 55773 + \frac{2007720}{x + 36}$

Simplify the quotient polynomial.

$x^{3} - 43 x^{2} + 1551 x - 55773 + \frac{2007720}{x + 36}$

Step 27

Since $- 36 < 0$ and the signs in the bottom row of the synthetic division alternate sign, $- 36$ is a lower bound for the real roots of the function.

Lower Bound: $- 36$

Step 28

Apply synthetic division on $\frac{x^{4} - 7 x^{3} + 3 x^{2} + 63 x - 108}{x - 54}$ when $x = 54$ .

Place the numbers representing the divisor and the dividend into a division-like configuration.

$54$	$1$	$- 7$	$3$	$63$	$- 108$
	\xa0				\xa0

The first number in the dividend $(1)$ is put into the first position of the result area (below the horizontal line).

$54$	$1$	$- 7$	$3$	$63$	$- 108$
	\xa0				\xa0
	$1$

Multiply the newest entry in the result $(1)$ by the divisor $(54)$ and place the result of $(54)$ under the next term in the dividend $(- 7)$ .

$54$	$1$	$- 7$	$3$	$63$	$- 108$
	\xa0	$54$			\xa0
	$1$

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$54$	$1$	$- 7$	$3$	$63$	$- 108$
	\xa0	$54$			\xa0
	$1$	$47$

Multiply the newest entry in the result $(47)$ by the divisor $(54)$ and place the result of $(2538)$ under the next term in the dividend $(3)$ .

$54$	$1$	$- 7$	$3$	$63$	$- 108$
	\xa0	$54$	$2538$		\xa0
	$1$	$47$

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$54$	$1$	$- 7$	$3$	$63$	$- 108$
	\xa0	$54$	$2538$		\xa0
	$1$	$47$	$2541$

Multiply the newest entry in the result $(2541)$ by the divisor $(54)$ and place the result of $(137214)$ under the next term in the dividend $(63)$ .

$54$	$1$	$- 7$	$3$	$63$	$- 108$
	\xa0	$54$	$2538$	$137214$	\xa0
	$1$	$47$	$2541$

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$54$	$1$	$- 7$	$3$	$63$	$- 108$
	\xa0	$54$	$2538$	$137214$	\xa0
	$1$	$47$	$2541$	$137277$

Multiply the newest entry in the result $(137277)$ by the divisor $(54)$ and place the result of $(7412958)$ under the next term in the dividend $(- 108)$ .

$54$	$1$	$- 7$	$3$	$63$	$- 108$
	\xa0	$54$	$2538$	$137214$	$7412958$
	$1$	$47$	$2541$	$137277$

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$54$	$1$	$- 7$	$3$	$63$	$- 108$
	\xa0	$54$	$2538$	$137214$	$7412958$
	$1$	$47$	$2541$	$137277$	$7412850$

All numbers except the last become the coefficients of the quotient polynomial. The last value in the result line is the remainder.

$1 x^{3} + 47 x^{2} + (2541) x + 137277 + \frac{7412850}{x - 54}$

Simplify the quotient polynomial.

$x^{3} + 47 x^{2} + 2541 x + 137277 + \frac{7412850}{x - 54}$

Step 29

Since $54 > 0$ and all of the signs in the bottom row of the synthetic division are positive, $54$ is an upper bound for the real roots of the function.

Upper Bound: $54$

Step 30

Apply synthetic division on $\frac{x^{4} - 7 x^{3} + 3 x^{2} + 63 x - 108}{x + 54}$ when $x = - 54$ .

Place the numbers representing the divisor and the dividend into a division-like configuration.

$- 54$	$1$	$- 7$	$3$	$63$	$- 108$
	\xa0				\xa0

The first number in the dividend $(1)$ is put into the first position of the result area (below the horizontal line).

$- 54$	$1$	$- 7$	$3$	$63$	$- 108$
	\xa0				\xa0
	$1$

Multiply the newest entry in the result $(1)$ by the divisor $(- 54)$ and place the result of $(- 54)$ under the next term in the dividend $(- 7)$ .

$- 54$	$1$	$- 7$	$3$	$63$	$- 108$
	\xa0	$- 54$			\xa0
	$1$

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$- 54$	$1$	$- 7$	$3$	$63$	$- 108$
	\xa0	$- 54$			\xa0
	$1$	$- 61$

Multiply the newest entry in the result $(- 61)$ by the divisor $(- 54)$ and place the result of $(3294)$ under the next term in the dividend $(3)$ .

$- 54$	$1$	$- 7$	$3$	$63$	$- 108$
	\xa0	$- 54$	$3294$		\xa0
	$1$	$- 61$

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$- 54$	$1$	$- 7$	$3$	$63$	$- 108$
	\xa0	$- 54$	$3294$		\xa0
	$1$	$- 61$	$3297$

Multiply the newest entry in the result $(3297)$ by the divisor $(- 54)$ and place the result of $(- 178038)$ under the next term in the dividend $(63)$ .

$- 54$	$1$	$- 7$	$3$	$63$	$- 108$
	\xa0	$- 54$	$3294$	$- 178038$	\xa0
	$1$	$- 61$	$3297$

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$- 54$	$1$	$- 7$	$3$	$63$	$- 108$
	\xa0	$- 54$	$3294$	$- 178038$	\xa0
	$1$	$- 61$	$3297$	$- 177975$

Multiply the newest entry in the result $(- 177975)$ by the divisor $(- 54)$ and place the result of $(9610650)$ under the next term in the dividend $(- 108)$ .

$- 54$	$1$	$- 7$	$3$	$63$	$- 108$
	\xa0	$- 54$	$3294$	$- 178038$	$9610650$
	$1$	$- 61$	$3297$	$- 177975$

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$- 54$	$1$	$- 7$	$3$	$63$	$- 108$
	\xa0	$- 54$	$3294$	$- 178038$	$9610650$
	$1$	$- 61$	$3297$	$- 177975$	$9610542$

All numbers except the last become the coefficients of the quotient polynomial. The last value in the result line is the remainder.

$1 x^{3} + - 61 x^{2} + (3297) x - 177975 + \frac{9610542}{x + 54}$

Simplify the quotient polynomial.

$x^{3} - 61 x^{2} + 3297 x - 177975 + \frac{9610542}{x + 54}$

Step 31

Since $- 54 < 0$ and the signs in the bottom row of the synthetic division alternate sign, $- 54$ is a lower bound for the real roots of the function.

Lower Bound: $- 54$

Step 32

Apply synthetic division on $\frac{x^{4} - 7 x^{3} + 3 x^{2} + 63 x - 108}{x - 108}$ when $x = 108$ .

Place the numbers representing the divisor and the dividend into a division-like configuration.

$108$	$1$	$- 7$	$3$	$63$	$- 108$
	\xa0				\xa0

The first number in the dividend $(1)$ is put into the first position of the result area (below the horizontal line).

$108$	$1$	$- 7$	$3$	$63$	$- 108$
	\xa0				\xa0
	$1$

Multiply the newest entry in the result $(1)$ by the divisor $(108)$ and place the result of $(108)$ under the next term in the dividend $(- 7)$ .

$108$	$1$	$- 7$	$3$	$63$	$- 108$
	\xa0	$108$			\xa0
	$1$

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$108$	$1$	$- 7$	$3$	$63$	$- 108$
	\xa0	$108$			\xa0
	$1$	$101$

Multiply the newest entry in the result $(101)$ by the divisor $(108)$ and place the result of $(10908)$ under the next term in the dividend $(3)$ .

$108$	$1$	$- 7$	$3$	$63$	$- 108$
	\xa0	$108$	$10908$		\xa0
	$1$	$101$

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$108$	$1$	$- 7$	$3$	$63$	$- 108$
	\xa0	$108$	$10908$		\xa0
	$1$	$101$	$10911$

Multiply the newest entry in the result $(10911)$ by the divisor $(108)$ and place the result of $(1178388)$ under the next term in the dividend $(63)$ .

$108$	$1$	$- 7$	$3$	$63$	$- 108$
	\xa0	$108$	$10908$	$1178388$	\xa0
	$1$	$101$	$10911$

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$108$	$1$	$- 7$	$3$	$63$	$- 108$
	\xa0	$108$	$10908$	$1178388$	\xa0
	$1$	$101$	$10911$	$1178451$

Multiply the newest entry in the result $(1178451)$ by the divisor $(108)$ and place the result of $(127272708)$ under the next term in the dividend $(- 108)$ .

$108$	$1$	$- 7$	$3$	$63$	$- 108$
	\xa0	$108$	$10908$	$1178388$	$127272708$
	$1$	$101$	$10911$	$1178451$

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$108$	$1$	$- 7$	$3$	$63$	$- 108$
	\xa0	$108$	$10908$	$1178388$	$127272708$
	$1$	$101$	$10911$	$1178451$	$127272600$

All numbers except the last become the coefficients of the quotient polynomial. The last value in the result line is the remainder.

$1 x^{3} + 101 x^{2} + (10911) x + 1178451 + \frac{127272600}{x - 108}$

Simplify the quotient polynomial.

$x^{3} + 101 x^{2} + 10911 x + 1178451 + \frac{127272600}{x - 108}$

Step 33

Since $108 > 0$ and all of the signs in the bottom row of the synthetic division are positive, $108$ is an upper bound for the real roots of the function.

Upper Bound: $108$

Step 34

Apply synthetic division on $\frac{x^{4} - 7 x^{3} + 3 x^{2} + 63 x - 108}{x + 108}$ when $x = - 108$ .

Place the numbers representing the divisor and the dividend into a division-like configuration.

$- 108$	$1$	$- 7$	$3$	$63$	$- 108$
	\xa0				\xa0

The first number in the dividend $(1)$ is put into the first position of the result area (below the horizontal line).

$- 108$	$1$	$- 7$	$3$	$63$	$- 108$
	\xa0				\xa0
	$1$

Multiply the newest entry in the result $(1)$ by the divisor $(- 108)$ and place the result of $(- 108)$ under the next term in the dividend $(- 7)$ .

$- 108$	$1$	$- 7$	$3$	$63$	$- 108$
	\xa0	$- 108$			\xa0
	$1$

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$- 108$	$1$	$- 7$	$3$	$63$	$- 108$
	\xa0	$- 108$			\xa0
	$1$	$- 115$

Multiply the newest entry in the result $(- 115)$ by the divisor $(- 108)$ and place the result of $(12420)$ under the next term in the dividend $(3)$ .

$- 108$	$1$	$- 7$	$3$	$63$	$- 108$
	\xa0	$- 108$	$12420$		\xa0
	$1$	$- 115$

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$- 108$	$1$	$- 7$	$3$	$63$	$- 108$
	\xa0	$- 108$	$12420$		\xa0
	$1$	$- 115$	$12423$

Multiply the newest entry in the result $(12423)$ by the divisor $(- 108)$ and place the result of $(- 1341684)$ under the next term in the dividend $(63)$ .

$- 108$	$1$	$- 7$	$3$	$63$	$- 108$
	\xa0	$- 108$	$12420$	$- 1341684$	\xa0
	$1$	$- 115$	$12423$

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$- 108$	$1$	$- 7$	$3$	$63$	$- 108$
	\xa0	$- 108$	$12420$	$- 1341684$	\xa0
	$1$	$- 115$	$12423$	$- 1341621$

Multiply the newest entry in the result $(- 1341621)$ by the divisor $(- 108)$ and place the result of $(144895068)$ under the next term in the dividend $(- 108)$ .

$- 108$	$1$	$- 7$	$3$	$63$	$- 108$
	\xa0	$- 108$	$12420$	$- 1341684$	$144895068$
	$1$	$- 115$	$12423$	$- 1341621$

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$- 108$	$1$	$- 7$	$3$	$63$	$- 108$
	\xa0	$- 108$	$12420$	$- 1341684$	$144895068$
	$1$	$- 115$	$12423$	$- 1341621$	$144894960$

All numbers except the last become the coefficients of the quotient polynomial. The last value in the result line is the remainder.

$1 x^{3} + - 115 x^{2} + (12423) x - 1341621 + \frac{144894960}{x + 108}$

Simplify the quotient polynomial.

$x^{3} - 115 x^{2} + 12423 x - 1341621 + \frac{144894960}{x + 108}$

Step 35

Since $- 108 < 0$ and the signs in the bottom row of the synthetic division alternate sign, $- 108$ is a lower bound for the real roots of the function.

Lower Bound: $- 108$

Step 36

Determine the upper and lower bounds.

Upper Bounds: $9, 12, 18, 27, 36, 54, 108$

Lower Bounds: $- 3, - 4, - 6, - 9, - 12, - 18, - 27, - 36, - 54, - 108$

Step 37

Find the Upper and Lower Bounds x^4-7x^3+3x^2+63x-108

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