Find the Upper and Lower Bounds h(x)=x^4-7x^3+8x^2+28x-48

$h (x) = x^{4} - 7 x^{3} + 8 x^{2} + 28 x - 48$

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.

$p = \pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 8, \pm 12, \pm 16, \pm 24, \pm 48$

$q = \pm 1$

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

$\pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 8, \pm 12, \pm 16, \pm 24, \pm 48$

Step 2

Apply synthetic division on $\frac{x^{4} - 7 x^{3} + 8 x^{2} + 28 x - 48}{x + 2}$ when $x = - 2$ .

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

$- 2$	$1$	$- 7$	$8$	$28$	$- 48$
	\xa0				\xa0

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

$- 2$	$1$	$- 7$	$8$	$28$	$- 48$
	\xa0				\xa0
	$1$

Multiply the newest entry in the result $(1)$ by the divisor $(- 2)$ and place the result of $(- 2)$ under the next term in the dividend $(- 7)$ .

$- 2$	$1$	$- 7$	$8$	$28$	$- 48$
	\xa0	$- 2$			\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.

$- 2$	$1$	$- 7$	$8$	$28$	$- 48$
	\xa0	$- 2$			\xa0
	$1$	$- 9$

Multiply the newest entry in the result $(- 9)$ by the divisor $(- 2)$ and place the result of $(18)$ under the next term in the dividend $(8)$ .

$- 2$	$1$	$- 7$	$8$	$28$	$- 48$
	\xa0	$- 2$	$18$		\xa0
	$1$	$- 9$

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

$- 2$	$1$	$- 7$	$8$	$28$	$- 48$
	\xa0	$- 2$	$18$		\xa0
	$1$	$- 9$	$26$

Multiply the newest entry in the result $(26)$ by the divisor $(- 2)$ and place the result of $(- 52)$ under the next term in the dividend $(28)$ .

$- 2$	$1$	$- 7$	$8$	$28$	$- 48$
	\xa0	$- 2$	$18$	$- 52$	\xa0
	$1$	$- 9$	$26$

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

$- 2$	$1$	$- 7$	$8$	$28$	$- 48$
	\xa0	$- 2$	$18$	$- 52$	\xa0
	$1$	$- 9$	$26$	$- 24$

Multiply the newest entry in the result $(- 24)$ by the divisor $(- 2)$ and place the result of $(48)$ under the next term in the dividend $(- 48)$ .

$- 2$	$1$	$- 7$	$8$	$28$	$- 48$
	\xa0	$- 2$	$18$	$- 52$	$48$
	$1$	$- 9$	$26$	$- 24$

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

$- 2$	$1$	$- 7$	$8$	$28$	$- 48$
	\xa0	$- 2$	$18$	$- 52$	$48$
	$1$	$- 9$	$26$	$- 24$	$0$

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

$1 x^{3} + - 9 x^{2} + (26) x - 24$

Simplify the quotient polynomial.

$x^{3} - 9 x^{2} + 26 x - 24$

Step 3

Since $- 2 < 0$ and the signs in the bottom row of the synthetic division alternate sign, $- 2$ is a lower bound for the real roots of the function.

Lower Bound: $- 2$

Step 4

Apply synthetic division on $\frac{x^{4} - 7 x^{3} + 8 x^{2} + 28 x - 48}{x + 3}$ when $x = - 3$ .

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

$- 3$	$1$	$- 7$	$8$	$28$	$- 48$
	\xa0				\xa0

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

$- 3$	$1$	$- 7$	$8$	$28$	$- 48$
	\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 $(- 7)$ .

$- 3$	$1$	$- 7$	$8$	$28$	$- 48$
	\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.

$- 3$	$1$	$- 7$	$8$	$28$	$- 48$
	\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 $(8)$ .

$- 3$	$1$	$- 7$	$8$	$28$	$- 48$
	\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.

$- 3$	$1$	$- 7$	$8$	$28$	$- 48$
	\xa0	$- 3$	$30$		\xa0
	$1$	$- 10$	$38$

Multiply the newest entry in the result $(38)$ by the divisor $(- 3)$ and place the result of $(- 114)$ under the next term in the dividend $(28)$ .

$- 3$	$1$	$- 7$	$8$	$28$	$- 48$
	\xa0	$- 3$	$30$	$- 114$	\xa0
	$1$	$- 10$	$38$

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

$- 3$	$1$	$- 7$	$8$	$28$	$- 48$
	\xa0	$- 3$	$30$	$- 114$	\xa0
	$1$	$- 10$	$38$	$- 86$

Multiply the newest entry in the result $(- 86)$ by the divisor $(- 3)$ and place the result of $(258)$ under the next term in the dividend $(- 48)$ .

$- 3$	$1$	$- 7$	$8$	$28$	$- 48$
	\xa0	$- 3$	$30$	$- 114$	$258$
	$1$	$- 10$	$38$	$- 86$

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

$- 3$	$1$	$- 7$	$8$	$28$	$- 48$
	\xa0	$- 3$	$30$	$- 114$	$258$
	$1$	$- 10$	$38$	$- 86$	$210$

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

$1 x^{3} + - 10 x^{2} + (38) x - 86 + \frac{210}{x + 3}$

Simplify the quotient polynomial.

$x^{3} - 10 x^{2} + 38 x - 86 + \frac{210}{x + 3}$

Step 5

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 6

Apply synthetic division on $\frac{x^{4} - 7 x^{3} + 8 x^{2} + 28 x - 48}{x + 4}$ when $x = - 4$ .

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

$- 4$	$1$	$- 7$	$8$	$28$	$- 48$
	\xa0				\xa0

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

$- 4$	$1$	$- 7$	$8$	$28$	$- 48$
	\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 $(- 7)$ .

$- 4$	$1$	$- 7$	$8$	$28$	$- 48$
	\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.

$- 4$	$1$	$- 7$	$8$	$28$	$- 48$
	\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 $(8)$ .

$- 4$	$1$	$- 7$	$8$	$28$	$- 48$
	\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.

$- 4$	$1$	$- 7$	$8$	$28$	$- 48$
	\xa0	$- 4$	$44$		\xa0
	$1$	$- 11$	$52$

Multiply the newest entry in the result $(52)$ by the divisor $(- 4)$ and place the result of $(- 208)$ under the next term in the dividend $(28)$ .

$- 4$	$1$	$- 7$	$8$	$28$	$- 48$
	\xa0	$- 4$	$44$	$- 208$	\xa0
	$1$	$- 11$	$52$

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

$- 4$	$1$	$- 7$	$8$	$28$	$- 48$
	\xa0	$- 4$	$44$	$- 208$	\xa0
	$1$	$- 11$	$52$	$- 180$

Multiply the newest entry in the result $(- 180)$ by the divisor $(- 4)$ and place the result of $(720)$ under the next term in the dividend $(- 48)$ .

$- 4$	$1$	$- 7$	$8$	$28$	$- 48$
	\xa0	$- 4$	$44$	$- 208$	$720$
	$1$	$- 11$	$52$	$- 180$

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

$- 4$	$1$	$- 7$	$8$	$28$	$- 48$
	\xa0	$- 4$	$44$	$- 208$	$720$
	$1$	$- 11$	$52$	$- 180$	$672$

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

$1 x^{3} + - 11 x^{2} + (52) x - 180 + \frac{672}{x + 4}$

Simplify the quotient polynomial.

$x^{3} - 11 x^{2} + 52 x - 180 + \frac{672}{x + 4}$

Step 7

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 8

Apply synthetic division on $\frac{x^{4} - 7 x^{3} + 8 x^{2} + 28 x - 48}{x + 6}$ when $x = - 6$ .

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

$- 6$	$1$	$- 7$	$8$	$28$	$- 48$
	\xa0				\xa0

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

$- 6$	$1$	$- 7$	$8$	$28$	$- 48$
	\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 $(- 7)$ .

$- 6$	$1$	$- 7$	$8$	$28$	$- 48$
	\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.

$- 6$	$1$	$- 7$	$8$	$28$	$- 48$
	\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 $(8)$ .

$- 6$	$1$	$- 7$	$8$	$28$	$- 48$
	\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.

$- 6$	$1$	$- 7$	$8$	$28$	$- 48$
	\xa0	$- 6$	$78$		\xa0
	$1$	$- 13$	$86$

Multiply the newest entry in the result $(86)$ by the divisor $(- 6)$ and place the result of $(- 516)$ under the next term in the dividend $(28)$ .

$- 6$	$1$	$- 7$	$8$	$28$	$- 48$
	\xa0	$- 6$	$78$	$- 516$	\xa0
	$1$	$- 13$	$86$

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

$- 6$	$1$	$- 7$	$8$	$28$	$- 48$
	\xa0	$- 6$	$78$	$- 516$	\xa0
	$1$	$- 13$	$86$	$- 488$

Multiply the newest entry in the result $(- 488)$ by the divisor $(- 6)$ and place the result of $(2928)$ under the next term in the dividend $(- 48)$ .

$- 6$	$1$	$- 7$	$8$	$28$	$- 48$
	\xa0	$- 6$	$78$	$- 516$	$2928$
	$1$	$- 13$	$86$	$- 488$

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

$- 6$	$1$	$- 7$	$8$	$28$	$- 48$
	\xa0	$- 6$	$78$	$- 516$	$2928$
	$1$	$- 13$	$86$	$- 488$	$2880$

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

$1 x^{3} + - 13 x^{2} + (86) x - 488 + \frac{2880}{x + 6}$

Simplify the quotient polynomial.

$x^{3} - 13 x^{2} + 86 x - 488 + \frac{2880}{x + 6}$

Step 9

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 10

Apply synthetic division on $\frac{x^{4} - 7 x^{3} + 8 x^{2} + 28 x - 48}{x - 8}$ when $x = 8$ .

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

$8$	$1$	$- 7$	$8$	$28$	$- 48$
	\xa0				\xa0

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

$8$	$1$	$- 7$	$8$	$28$	$- 48$
	\xa0				\xa0
	$1$

Multiply the newest entry in the result $(1)$ by the divisor $(8)$ and place the result of $(8)$ under the next term in the dividend $(- 7)$ .

$8$	$1$	$- 7$	$8$	$28$	$- 48$
	\xa0	$8$			\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.

$8$	$1$	$- 7$	$8$	$28$	$- 48$
	\xa0	$8$			\xa0
	$1$	$1$

Multiply the newest entry in the result $(1)$ by the divisor $(8)$ and place the result of $(8)$ under the next term in the dividend $(8)$ .

$8$	$1$	$- 7$	$8$	$28$	$- 48$
	\xa0	$8$	$8$		\xa0
	$1$	$1$

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

$8$	$1$	$- 7$	$8$	$28$	$- 48$
	\xa0	$8$	$8$		\xa0
	$1$	$1$	$16$

Multiply the newest entry in the result $(16)$ by the divisor $(8)$ and place the result of $(128)$ under the next term in the dividend $(28)$ .

$8$	$1$	$- 7$	$8$	$28$	$- 48$
	\xa0	$8$	$8$	$128$	\xa0
	$1$	$1$	$16$

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

$8$	$1$	$- 7$	$8$	$28$	$- 48$
	\xa0	$8$	$8$	$128$	\xa0
	$1$	$1$	$16$	$156$

Multiply the newest entry in the result $(156)$ by the divisor $(8)$ and place the result of $(1248)$ under the next term in the dividend $(- 48)$ .

$8$	$1$	$- 7$	$8$	$28$	$- 48$
	\xa0	$8$	$8$	$128$	$1248$
	$1$	$1$	$16$	$156$

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

$8$	$1$	$- 7$	$8$	$28$	$- 48$
	\xa0	$8$	$8$	$128$	$1248$
	$1$	$1$	$16$	$156$	$1200$

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

$1 x^{3} + 1 x^{2} + (16) x + 156 + \frac{1200}{x - 8}$

Simplify the quotient polynomial.

$x^{3} + x^{2} + 16 x + 156 + \frac{1200}{x - 8}$

Step 11

Since $8 > 0$ and all of the signs in the bottom row of the synthetic division are positive, $8$ is an upper bound for the real roots of the function.

Upper Bound: $8$

Step 12

Apply synthetic division on $\frac{x^{4} - 7 x^{3} + 8 x^{2} + 28 x - 48}{x + 8}$ when $x = - 8$ .

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

$- 8$	$1$	$- 7$	$8$	$28$	$- 48$
	\xa0				\xa0

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

$- 8$	$1$	$- 7$	$8$	$28$	$- 48$
	\xa0				\xa0
	$1$

Multiply the newest entry in the result $(1)$ by the divisor $(- 8)$ and place the result of $(- 8)$ under the next term in the dividend $(- 7)$ .

$- 8$	$1$	$- 7$	$8$	$28$	$- 48$
	\xa0	$- 8$			\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.

$- 8$	$1$	$- 7$	$8$	$28$	$- 48$
	\xa0	$- 8$			\xa0
	$1$	$- 15$

Multiply the newest entry in the result $(- 15)$ by the divisor $(- 8)$ and place the result of $(120)$ under the next term in the dividend $(8)$ .

$- 8$	$1$	$- 7$	$8$	$28$	$- 48$
	\xa0	$- 8$	$120$		\xa0
	$1$	$- 15$

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

$- 8$	$1$	$- 7$	$8$	$28$	$- 48$
	\xa0	$- 8$	$120$		\xa0
	$1$	$- 15$	$128$

Multiply the newest entry in the result $(128)$ by the divisor $(- 8)$ and place the result of $(- 1024)$ under the next term in the dividend $(28)$ .

$- 8$	$1$	$- 7$	$8$	$28$	$- 48$
	\xa0	$- 8$	$120$	$- 1024$	\xa0
	$1$	$- 15$	$128$

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

$- 8$	$1$	$- 7$	$8$	$28$	$- 48$
	\xa0	$- 8$	$120$	$- 1024$	\xa0
	$1$	$- 15$	$128$	$- 996$

Multiply the newest entry in the result $(- 996)$ by the divisor $(- 8)$ and place the result of $(7968)$ under the next term in the dividend $(- 48)$ .

$- 8$	$1$	$- 7$	$8$	$28$	$- 48$
	\xa0	$- 8$	$120$	$- 1024$	$7968$
	$1$	$- 15$	$128$	$- 996$

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

$- 8$	$1$	$- 7$	$8$	$28$	$- 48$
	\xa0	$- 8$	$120$	$- 1024$	$7968$
	$1$	$- 15$	$128$	$- 996$	$7920$

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

$1 x^{3} + - 15 x^{2} + (128) x - 996 + \frac{7920}{x + 8}$

Simplify the quotient polynomial.

$x^{3} - 15 x^{2} + 128 x - 996 + \frac{7920}{x + 8}$

Step 13

Since $- 8 < 0$ and the signs in the bottom row of the synthetic division alternate sign, $- 8$ is a lower bound for the real roots of the function.

Lower Bound: $- 8$

Step 14

Apply synthetic division on $\frac{x^{4} - 7 x^{3} + 8 x^{2} + 28 x - 48}{x - 12}$ when $x = 12$ .

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

$12$	$1$	$- 7$	$8$	$28$	$- 48$
	\xa0				\xa0

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

$12$	$1$	$- 7$	$8$	$28$	$- 48$
	\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 $(- 7)$ .

$12$	$1$	$- 7$	$8$	$28$	$- 48$
	\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.

$12$	$1$	$- 7$	$8$	$28$	$- 48$
	\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 $(8)$ .

$12$	$1$	$- 7$	$8$	$28$	$- 48$
	\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.

$12$	$1$	$- 7$	$8$	$28$	$- 48$
	\xa0	$12$	$60$		\xa0
	$1$	$5$	$68$

Multiply the newest entry in the result $(68)$ by the divisor $(12)$ and place the result of $(816)$ under the next term in the dividend $(28)$ .

$12$	$1$	$- 7$	$8$	$28$	$- 48$
	\xa0	$12$	$60$	$816$	\xa0
	$1$	$5$	$68$

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

$12$	$1$	$- 7$	$8$	$28$	$- 48$
	\xa0	$12$	$60$	$816$	\xa0
	$1$	$5$	$68$	$844$

Multiply the newest entry in the result $(844)$ by the divisor $(12)$ and place the result of $(10128)$ under the next term in the dividend $(- 48)$ .

$12$	$1$	$- 7$	$8$	$28$	$- 48$
	\xa0	$12$	$60$	$816$	$10128$
	$1$	$5$	$68$	$844$

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

$12$	$1$	$- 7$	$8$	$28$	$- 48$
	\xa0	$12$	$60$	$816$	$10128$
	$1$	$5$	$68$	$844$	$10080$

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

$1 x^{3} + 5 x^{2} + (68) x + 844 + \frac{10080}{x - 12}$

Simplify the quotient polynomial.

$x^{3} + 5 x^{2} + 68 x + 844 + \frac{10080}{x - 12}$

Step 15

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 16

Apply synthetic division on $\frac{x^{4} - 7 x^{3} + 8 x^{2} + 28 x - 48}{x + 12}$ when $x = - 12$ .

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

$- 12$	$1$	$- 7$	$8$	$28$	$- 48$
	\xa0				\xa0

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

$- 12$	$1$	$- 7$	$8$	$28$	$- 48$
	\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 $(- 7)$ .

$- 12$	$1$	$- 7$	$8$	$28$	$- 48$
	\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.

$- 12$	$1$	$- 7$	$8$	$28$	$- 48$
	\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 $(8)$ .

$- 12$	$1$	$- 7$	$8$	$28$	$- 48$
	\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.

$- 12$	$1$	$- 7$	$8$	$28$	$- 48$
	\xa0	$- 12$	$228$		\xa0
	$1$	$- 19$	$236$

Multiply the newest entry in the result $(236)$ by the divisor $(- 12)$ and place the result of $(- 2832)$ under the next term in the dividend $(28)$ .

$- 12$	$1$	$- 7$	$8$	$28$	$- 48$
	\xa0	$- 12$	$228$	$- 2832$	\xa0
	$1$	$- 19$	$236$

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

$- 12$	$1$	$- 7$	$8$	$28$	$- 48$
	\xa0	$- 12$	$228$	$- 2832$	\xa0
	$1$	$- 19$	$236$	$- 2804$

Multiply the newest entry in the result $(- 2804)$ by the divisor $(- 12)$ and place the result of $(33648)$ under the next term in the dividend $(- 48)$ .

$- 12$	$1$	$- 7$	$8$	$28$	$- 48$
	\xa0	$- 12$	$228$	$- 2832$	$33648$
	$1$	$- 19$	$236$	$- 2804$

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

$- 12$	$1$	$- 7$	$8$	$28$	$- 48$
	\xa0	$- 12$	$228$	$- 2832$	$33648$
	$1$	$- 19$	$236$	$- 2804$	$33600$

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

$1 x^{3} + - 19 x^{2} + (236) x - 2804 + \frac{33600}{x + 12}$

Simplify the quotient polynomial.

$x^{3} - 19 x^{2} + 236 x - 2804 + \frac{33600}{x + 12}$

Step 17

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 18

Apply synthetic division on $\frac{x^{4} - 7 x^{3} + 8 x^{2} + 28 x - 48}{x - 16}$ when $x = 16$ .

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

$16$	$1$	$- 7$	$8$	$28$	$- 48$
	\xa0				\xa0

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

$16$	$1$	$- 7$	$8$	$28$	$- 48$
	\xa0				\xa0
	$1$

Multiply the newest entry in the result $(1)$ by the divisor $(16)$ and place the result of $(16)$ under the next term in the dividend $(- 7)$ .

$16$	$1$	$- 7$	$8$	$28$	$- 48$
	\xa0	$16$			\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.

$16$	$1$	$- 7$	$8$	$28$	$- 48$
	\xa0	$16$			\xa0
	$1$	$9$

Multiply the newest entry in the result $(9)$ by the divisor $(16)$ and place the result of $(144)$ under the next term in the dividend $(8)$ .

$16$	$1$	$- 7$	$8$	$28$	$- 48$
	\xa0	$16$	$144$		\xa0
	$1$	$9$

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

$16$	$1$	$- 7$	$8$	$28$	$- 48$
	\xa0	$16$	$144$		\xa0
	$1$	$9$	$152$

Multiply the newest entry in the result $(152)$ by the divisor $(16)$ and place the result of $(2432)$ under the next term in the dividend $(28)$ .

$16$	$1$	$- 7$	$8$	$28$	$- 48$
	\xa0	$16$	$144$	$2432$	\xa0
	$1$	$9$	$152$

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

$16$	$1$	$- 7$	$8$	$28$	$- 48$
	\xa0	$16$	$144$	$2432$	\xa0
	$1$	$9$	$152$	$2460$

Multiply the newest entry in the result $(2460)$ by the divisor $(16)$ and place the result of $(39360)$ under the next term in the dividend $(- 48)$ .

$16$	$1$	$- 7$	$8$	$28$	$- 48$
	\xa0	$16$	$144$	$2432$	$39360$
	$1$	$9$	$152$	$2460$

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

$16$	$1$	$- 7$	$8$	$28$	$- 48$
	\xa0	$16$	$144$	$2432$	$39360$
	$1$	$9$	$152$	$2460$	$39312$

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

$1 x^{3} + 9 x^{2} + (152) x + 2460 + \frac{39312}{x - 16}$

Simplify the quotient polynomial.

$x^{3} + 9 x^{2} + 152 x + 2460 + \frac{39312}{x - 16}$

Step 19

Since $16 > 0$ and all of the signs in the bottom row of the synthetic division are positive, $16$ is an upper bound for the real roots of the function.

Upper Bound: $16$

Step 20

Apply synthetic division on $\frac{x^{4} - 7 x^{3} + 8 x^{2} + 28 x - 48}{x + 16}$ when $x = - 16$ .

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

$- 16$	$1$	$- 7$	$8$	$28$	$- 48$
	\xa0				\xa0

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

$- 16$	$1$	$- 7$	$8$	$28$	$- 48$
	\xa0				\xa0
	$1$

Multiply the newest entry in the result $(1)$ by the divisor $(- 16)$ and place the result of $(- 16)$ under the next term in the dividend $(- 7)$ .

$- 16$	$1$	$- 7$	$8$	$28$	$- 48$
	\xa0	$- 16$			\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.

$- 16$	$1$	$- 7$	$8$	$28$	$- 48$
	\xa0	$- 16$			\xa0
	$1$	$- 23$

Multiply the newest entry in the result $(- 23)$ by the divisor $(- 16)$ and place the result of $(368)$ under the next term in the dividend $(8)$ .

$- 16$	$1$	$- 7$	$8$	$28$	$- 48$
	\xa0	$- 16$	$368$		\xa0
	$1$	$- 23$

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

$- 16$	$1$	$- 7$	$8$	$28$	$- 48$
	\xa0	$- 16$	$368$		\xa0
	$1$	$- 23$	$376$

Multiply the newest entry in the result $(376)$ by the divisor $(- 16)$ and place the result of $(- 6016)$ under the next term in the dividend $(28)$ .

$- 16$	$1$	$- 7$	$8$	$28$	$- 48$
	\xa0	$- 16$	$368$	$- 6016$	\xa0
	$1$	$- 23$	$376$

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

$- 16$	$1$	$- 7$	$8$	$28$	$- 48$
	\xa0	$- 16$	$368$	$- 6016$	\xa0
	$1$	$- 23$	$376$	$- 5988$

Multiply the newest entry in the result $(- 5988)$ by the divisor $(- 16)$ and place the result of $(95808)$ under the next term in the dividend $(- 48)$ .

$- 16$	$1$	$- 7$	$8$	$28$	$- 48$
	\xa0	$- 16$	$368$	$- 6016$	$95808$
	$1$	$- 23$	$376$	$- 5988$

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

$- 16$	$1$	$- 7$	$8$	$28$	$- 48$
	\xa0	$- 16$	$368$	$- 6016$	$95808$
	$1$	$- 23$	$376$	$- 5988$	$95760$

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

$1 x^{3} + - 23 x^{2} + (376) x - 5988 + \frac{95760}{x + 16}$

Simplify the quotient polynomial.

$x^{3} - 23 x^{2} + 376 x - 5988 + \frac{95760}{x + 16}$

Step 21

Since $- 16 < 0$ and the signs in the bottom row of the synthetic division alternate sign, $- 16$ is a lower bound for the real roots of the function.

Lower Bound: $- 16$

Step 22

Apply synthetic division on $\frac{x^{4} - 7 x^{3} + 8 x^{2} + 28 x - 48}{x - 24}$ when $x = 24$ .

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

$24$	$1$	$- 7$	$8$	$28$	$- 48$
	\xa0				\xa0

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

$24$	$1$	$- 7$	$8$	$28$	$- 48$
	\xa0				\xa0
	$1$

Multiply the newest entry in the result $(1)$ by the divisor $(24)$ and place the result of $(24)$ under the next term in the dividend $(- 7)$ .

$24$	$1$	$- 7$	$8$	$28$	$- 48$
	\xa0	$24$			\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.

$24$	$1$	$- 7$	$8$	$28$	$- 48$
	\xa0	$24$			\xa0
	$1$	$17$

Multiply the newest entry in the result $(17)$ by the divisor $(24)$ and place the result of $(408)$ under the next term in the dividend $(8)$ .

$24$	$1$	$- 7$	$8$	$28$	$- 48$
	\xa0	$24$	$408$		\xa0
	$1$	$17$

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

$24$	$1$	$- 7$	$8$	$28$	$- 48$
	\xa0	$24$	$408$		\xa0
	$1$	$17$	$416$

Multiply the newest entry in the result $(416)$ by the divisor $(24)$ and place the result of $(9984)$ under the next term in the dividend $(28)$ .

$24$	$1$	$- 7$	$8$	$28$	$- 48$
	\xa0	$24$	$408$	$9984$	\xa0
	$1$	$17$	$416$

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

$24$	$1$	$- 7$	$8$	$28$	$- 48$
	\xa0	$24$	$408$	$9984$	\xa0
	$1$	$17$	$416$	$10012$

Multiply the newest entry in the result $(10012)$ by the divisor $(24)$ and place the result of $(240288)$ under the next term in the dividend $(- 48)$ .

$24$	$1$	$- 7$	$8$	$28$	$- 48$
	\xa0	$24$	$408$	$9984$	$240288$
	$1$	$17$	$416$	$10012$

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

$24$	$1$	$- 7$	$8$	$28$	$- 48$
	\xa0	$24$	$408$	$9984$	$240288$
	$1$	$17$	$416$	$10012$	$240240$

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

$1 x^{3} + 17 x^{2} + (416) x + 10012 + \frac{240240}{x - 24}$

Simplify the quotient polynomial.

$x^{3} + 17 x^{2} + 416 x + 10012 + \frac{240240}{x - 24}$

Step 23

Since $24 > 0$ and all of the signs in the bottom row of the synthetic division are positive, $24$ is an upper bound for the real roots of the function.

Upper Bound: $24$

Step 24

Apply synthetic division on $\frac{x^{4} - 7 x^{3} + 8 x^{2} + 28 x - 48}{x + 24}$ when $x = - 24$ .

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

$- 24$	$1$	$- 7$	$8$	$28$	$- 48$
	\xa0				\xa0

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

$- 24$	$1$	$- 7$	$8$	$28$	$- 48$
	\xa0				\xa0
	$1$

Multiply the newest entry in the result $(1)$ by the divisor $(- 24)$ and place the result of $(- 24)$ under the next term in the dividend $(- 7)$ .

$- 24$	$1$	$- 7$	$8$	$28$	$- 48$
	\xa0	$- 24$			\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.

$- 24$	$1$	$- 7$	$8$	$28$	$- 48$
	\xa0	$- 24$			\xa0
	$1$	$- 31$

Multiply the newest entry in the result $(- 31)$ by the divisor $(- 24)$ and place the result of $(744)$ under the next term in the dividend $(8)$ .

$- 24$	$1$	$- 7$	$8$	$28$	$- 48$
	\xa0	$- 24$	$744$		\xa0
	$1$	$- 31$

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

$- 24$	$1$	$- 7$	$8$	$28$	$- 48$
	\xa0	$- 24$	$744$		\xa0
	$1$	$- 31$	$752$

Multiply the newest entry in the result $(752)$ by the divisor $(- 24)$ and place the result of $(- 18048)$ under the next term in the dividend $(28)$ .

$- 24$	$1$	$- 7$	$8$	$28$	$- 48$
	\xa0	$- 24$	$744$	$- 18048$	\xa0
	$1$	$- 31$	$752$

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

$- 24$	$1$	$- 7$	$8$	$28$	$- 48$
	\xa0	$- 24$	$744$	$- 18048$	\xa0
	$1$	$- 31$	$752$	$- 18020$

Multiply the newest entry in the result $(- 18020)$ by the divisor $(- 24)$ and place the result of $(432480)$ under the next term in the dividend $(- 48)$ .

$- 24$	$1$	$- 7$	$8$	$28$	$- 48$
	\xa0	$- 24$	$744$	$- 18048$	$432480$
	$1$	$- 31$	$752$	$- 18020$

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

$- 24$	$1$	$- 7$	$8$	$28$	$- 48$
	\xa0	$- 24$	$744$	$- 18048$	$432480$
	$1$	$- 31$	$752$	$- 18020$	$432432$

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

$1 x^{3} + - 31 x^{2} + (752) x - 18020 + \frac{432432}{x + 24}$

Simplify the quotient polynomial.

$x^{3} - 31 x^{2} + 752 x - 18020 + \frac{432432}{x + 24}$

Step 25

Since $- 24 < 0$ and the signs in the bottom row of the synthetic division alternate sign, $- 24$ is a lower bound for the real roots of the function.

Lower Bound: $- 24$

Step 26

Apply synthetic division on $\frac{x^{4} - 7 x^{3} + 8 x^{2} + 28 x - 48}{x - 48}$ when $x = 48$ .

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

$48$	$1$	$- 7$	$8$	$28$	$- 48$
	\xa0				\xa0

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

$48$	$1$	$- 7$	$8$	$28$	$- 48$
	\xa0				\xa0
	$1$

Multiply the newest entry in the result $(1)$ by the divisor $(48)$ and place the result of $(48)$ under the next term in the dividend $(- 7)$ .

$48$	$1$	$- 7$	$8$	$28$	$- 48$
	\xa0	$48$			\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.

$48$	$1$	$- 7$	$8$	$28$	$- 48$
	\xa0	$48$			\xa0
	$1$	$41$

Multiply the newest entry in the result $(41)$ by the divisor $(48)$ and place the result of $(1968)$ under the next term in the dividend $(8)$ .

$48$	$1$	$- 7$	$8$	$28$	$- 48$
	\xa0	$48$	$1968$		\xa0
	$1$	$41$

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

$48$	$1$	$- 7$	$8$	$28$	$- 48$
	\xa0	$48$	$1968$		\xa0
	$1$	$41$	$1976$

Multiply the newest entry in the result $(1976)$ by the divisor $(48)$ and place the result of $(94848)$ under the next term in the dividend $(28)$ .

$48$	$1$	$- 7$	$8$	$28$	$- 48$
	\xa0	$48$	$1968$	$94848$	\xa0
	$1$	$41$	$1976$

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

$48$	$1$	$- 7$	$8$	$28$	$- 48$
	\xa0	$48$	$1968$	$94848$	\xa0
	$1$	$41$	$1976$	$94876$

Multiply the newest entry in the result $(94876)$ by the divisor $(48)$ and place the result of $(4554048)$ under the next term in the dividend $(- 48)$ .

$48$	$1$	$- 7$	$8$	$28$	$- 48$
	\xa0	$48$	$1968$	$94848$	$4554048$
	$1$	$41$	$1976$	$94876$

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

$48$	$1$	$- 7$	$8$	$28$	$- 48$
	\xa0	$48$	$1968$	$94848$	$4554048$
	$1$	$41$	$1976$	$94876$	$4554000$

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

$1 x^{3} + 41 x^{2} + (1976) x + 94876 + \frac{4554000}{x - 48}$

Simplify the quotient polynomial.

$x^{3} + 41 x^{2} + 1976 x + 94876 + \frac{4554000}{x - 48}$

Step 27

Since $48 > 0$ and all of the signs in the bottom row of the synthetic division are positive, $48$ is an upper bound for the real roots of the function.

Upper Bound: $48$

Step 28

Apply synthetic division on $\frac{x^{4} - 7 x^{3} + 8 x^{2} + 28 x - 48}{x + 48}$ when $x = - 48$ .

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

$- 48$	$1$	$- 7$	$8$	$28$	$- 48$
	\xa0				\xa0

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

$- 48$	$1$	$- 7$	$8$	$28$	$- 48$
	\xa0				\xa0
	$1$

Multiply the newest entry in the result $(1)$ by the divisor $(- 48)$ and place the result of $(- 48)$ under the next term in the dividend $(- 7)$ .

$- 48$	$1$	$- 7$	$8$	$28$	$- 48$
	\xa0	$- 48$			\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.

$- 48$	$1$	$- 7$	$8$	$28$	$- 48$
	\xa0	$- 48$			\xa0
	$1$	$- 55$

Multiply the newest entry in the result $(- 55)$ by the divisor $(- 48)$ and place the result of $(2640)$ under the next term in the dividend $(8)$ .

$- 48$	$1$	$- 7$	$8$	$28$	$- 48$
	\xa0	$- 48$	$2640$		\xa0
	$1$	$- 55$

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

$- 48$	$1$	$- 7$	$8$	$28$	$- 48$
	\xa0	$- 48$	$2640$		\xa0
	$1$	$- 55$	$2648$

Multiply the newest entry in the result $(2648)$ by the divisor $(- 48)$ and place the result of $(- 127104)$ under the next term in the dividend $(28)$ .

$- 48$	$1$	$- 7$	$8$	$28$	$- 48$
	\xa0	$- 48$	$2640$	$- 127104$	\xa0
	$1$	$- 55$	$2648$

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

$- 48$	$1$	$- 7$	$8$	$28$	$- 48$
	\xa0	$- 48$	$2640$	$- 127104$	\xa0
	$1$	$- 55$	$2648$	$- 127076$

Multiply the newest entry in the result $(- 127076)$ by the divisor $(- 48)$ and place the result of $(6099648)$ under the next term in the dividend $(- 48)$ .

$- 48$	$1$	$- 7$	$8$	$28$	$- 48$
	\xa0	$- 48$	$2640$	$- 127104$	$6099648$
	$1$	$- 55$	$2648$	$- 127076$

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

$- 48$	$1$	$- 7$	$8$	$28$	$- 48$
	\xa0	$- 48$	$2640$	$- 127104$	$6099648$
	$1$	$- 55$	$2648$	$- 127076$	$6099600$

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

$1 x^{3} + - 55 x^{2} + (2648) x - 127076 + \frac{6099600}{x + 48}$

Simplify the quotient polynomial.

$x^{3} - 55 x^{2} + 2648 x - 127076 + \frac{6099600}{x + 48}$

Step 29

Since $- 48 < 0$ and the signs in the bottom row of the synthetic division alternate sign, $- 48$ is a lower bound for the real roots of the function.

Lower Bound: $- 48$

Step 30

Determine the upper and lower bounds.

Upper Bounds: $8, 12, 16, 24, 48$

Lower Bounds: $- 2, - 3, - 4, - 6, - 8, - 12, - 16, - 24, - 48$

Step 31

Find the Upper and Lower Bounds h(x)=x^4-7x^3+8x^2+28x-48

Related Questions

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