Find the Upper and Lower Bounds f(x)=x^3+6x^2-3x-18

$f (x) = x^{3} + 6 x^{2} - 3 x - 18$

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.

$p = \pm 1, \pm 2, \pm 3, \pm 6, \pm 9, \pm 18$

$q = \pm 1$

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

$\pm 1, \pm 2, \pm 3, \pm 6, \pm 9, \pm 18$

Step 2

Apply synthetic division on $\frac{x^{3} + 6 x^{2} - 3 x - 18}{x - 2}$ when $x = 2$ .

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

$2$	$1$	$6$	$- 3$	$- 18$
	\xa0			\xa0

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

$2$	$1$	$6$	$- 3$	$- 18$
	\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 $(6)$ .

$2$	$1$	$6$	$- 3$	$- 18$
	\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.

$2$	$1$	$6$	$- 3$	$- 18$
	\xa0	$2$		\xa0
	$1$	$8$

Multiply the newest entry in the result $(8)$ by the divisor $(2)$ and place the result of $(16)$ under the next term in the dividend $(- 3)$ .

$2$	$1$	$6$	$- 3$	$- 18$
	\xa0	$2$	$16$	\xa0
	$1$	$8$

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

$2$	$1$	$6$	$- 3$	$- 18$
	\xa0	$2$	$16$	\xa0
	$1$	$8$	$13$

Multiply the newest entry in the result $(13)$ by the divisor $(2)$ and place the result of $(26)$ under the next term in the dividend $(- 18)$ .

$2$	$1$	$6$	$- 3$	$- 18$
	\xa0	$2$	$16$	$26$
	$1$	$8$	$13$

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

$2$	$1$	$6$	$- 3$	$- 18$
	\xa0	$2$	$16$	$26$
	$1$	$8$	$13$	$8$

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

$1 x^{2} + 8 x + 13 + \frac{8}{x - 2}$

Simplify the quotient polynomial.

$x^{2} + 8 x + 13 + \frac{8}{x - 2}$

Step 3

Since $2 > 0$ and all of the signs in the bottom row of the synthetic division are positive, $2$ is an upper bound for the real roots of the function.

Upper Bound: $2$

Step 4

Apply synthetic division on $\frac{x^{3} + 6 x^{2} - 3 x - 18}{x - 3}$ when $x = 3$ .

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

$3$	$1$	$6$	$- 3$	$- 18$
	\xa0			\xa0

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

$3$	$1$	$6$	$- 3$	$- 18$
	\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 $(6)$ .

$3$	$1$	$6$	$- 3$	$- 18$
	\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.

$3$	$1$	$6$	$- 3$	$- 18$
	\xa0	$3$		\xa0
	$1$	$9$

Multiply the newest entry in the result $(9)$ by the divisor $(3)$ and place the result of $(27)$ under the next term in the dividend $(- 3)$ .

$3$	$1$	$6$	$- 3$	$- 18$
	\xa0	$3$	$27$	\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.

$3$	$1$	$6$	$- 3$	$- 18$
	\xa0	$3$	$27$	\xa0
	$1$	$9$	$24$

Multiply the newest entry in the result $(24)$ by the divisor $(3)$ and place the result of $(72)$ under the next term in the dividend $(- 18)$ .

$3$	$1$	$6$	$- 3$	$- 18$
	\xa0	$3$	$27$	$72$
	$1$	$9$	$24$

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

$3$	$1$	$6$	$- 3$	$- 18$
	\xa0	$3$	$27$	$72$
	$1$	$9$	$24$	$54$

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

$1 x^{2} + 9 x + 24 + \frac{54}{x - 3}$

Simplify the quotient polynomial.

$x^{2} + 9 x + 24 + \frac{54}{x - 3}$

Step 5

Since $3 > 0$ and all of the signs in the bottom row of the synthetic division are positive, $3$ is an upper bound for the real roots of the function.

Upper Bound: $3$

Step 6

Apply synthetic division on $\frac{x^{3} + 6 x^{2} - 3 x - 18}{x - 6}$ when $x = 6$ .

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

$6$	$1$	$6$	$- 3$	$- 18$
	\xa0			\xa0

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

$6$	$1$	$6$	$- 3$	$- 18$
	\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 $(6)$ .

$6$	$1$	$6$	$- 3$	$- 18$
	\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.

$6$	$1$	$6$	$- 3$	$- 18$
	\xa0	$6$		\xa0
	$1$	$12$

Multiply the newest entry in the result $(12)$ by the divisor $(6)$ and place the result of $(72)$ under the next term in the dividend $(- 3)$ .

$6$	$1$	$6$	$- 3$	$- 18$
	\xa0	$6$	$72$	\xa0
	$1$	$12$

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

$6$	$1$	$6$	$- 3$	$- 18$
	\xa0	$6$	$72$	\xa0
	$1$	$12$	$69$

Multiply the newest entry in the result $(69)$ by the divisor $(6)$ and place the result of $(414)$ under the next term in the dividend $(- 18)$ .

$6$	$1$	$6$	$- 3$	$- 18$
	\xa0	$6$	$72$	$414$
	$1$	$12$	$69$

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

$6$	$1$	$6$	$- 3$	$- 18$
	\xa0	$6$	$72$	$414$
	$1$	$12$	$69$	$396$

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

$1 x^{2} + 12 x + 69 + \frac{396}{x - 6}$

Simplify the quotient polynomial.

$x^{2} + 12 x + 69 + \frac{396}{x - 6}$

Step 7

Since $6 > 0$ and all of the signs in the bottom row of the synthetic division are positive, $6$ is an upper bound for the real roots of the function.

Upper Bound: $6$

Step 8

Apply synthetic division on $\frac{x^{3} + 6 x^{2} - 3 x - 18}{x + 6}$ when $x = - 6$ .

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

$- 6$	$1$	$6$	$- 3$	$- 18$
	\xa0			\xa0

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

$- 6$	$1$	$6$	$- 3$	$- 18$
	\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 $(6)$ .

$- 6$	$1$	$6$	$- 3$	$- 18$
	\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.

$- 6$	$1$	$6$	$- 3$	$- 18$
	\xa0	$- 6$		\xa0
	$1$	$0$

Multiply the newest entry in the result $(0)$ by the divisor $(- 6)$ and place the result of $(0)$ under the next term in the dividend $(- 3)$ .

$- 6$	$1$	$6$	$- 3$	$- 18$
	\xa0	$- 6$	$0$	\xa0
	$1$	$0$

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

$- 6$	$1$	$6$	$- 3$	$- 18$
	\xa0	$- 6$	$0$	\xa0
	$1$	$0$	$- 3$

Multiply the newest entry in the result $(- 3)$ by the divisor $(- 6)$ and place the result of $(18)$ under the next term in the dividend $(- 18)$ .

$- 6$	$1$	$6$	$- 3$	$- 18$
	\xa0	$- 6$	$0$	$18$
	$1$	$0$	$- 3$

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

$- 6$	$1$	$6$	$- 3$	$- 18$
	\xa0	$- 6$	$0$	$18$
	$1$	$0$	$- 3$	$0$

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

$1 x^{2} + 0 x - 3$

Simplify the quotient polynomial.

$x^{2} - 3$

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 $\frac{x^{3} + 6 x^{2} - 3 x - 18}{x - 9}$ when $x = 9$ .

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

$9$	$1$	$6$	$- 3$	$- 18$
	\xa0			\xa0

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

$9$	$1$	$6$	$- 3$	$- 18$
	\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 $(6)$ .

$9$	$1$	$6$	$- 3$	$- 18$
	\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.

$9$	$1$	$6$	$- 3$	$- 18$
	\xa0	$9$		\xa0
	$1$	$15$

Multiply the newest entry in the result $(15)$ by the divisor $(9)$ and place the result of $(135)$ under the next term in the dividend $(- 3)$ .

$9$	$1$	$6$	$- 3$	$- 18$
	\xa0	$9$	$135$	\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.

$9$	$1$	$6$	$- 3$	$- 18$
	\xa0	$9$	$135$	\xa0
	$1$	$15$	$132$

Multiply the newest entry in the result $(132)$ by the divisor $(9)$ and place the result of $(1188)$ under the next term in the dividend $(- 18)$ .

$9$	$1$	$6$	$- 3$	$- 18$
	\xa0	$9$	$135$	$1188$
	$1$	$15$	$132$

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

$9$	$1$	$6$	$- 3$	$- 18$
	\xa0	$9$	$135$	$1188$
	$1$	$15$	$132$	$1170$

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

$1 x^{2} + 15 x + 132 + \frac{1170}{x - 9}$

Simplify the quotient polynomial.

$x^{2} + 15 x + 132 + \frac{1170}{x - 9}$

Step 11

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 12

Apply synthetic division on $\frac{x^{3} + 6 x^{2} - 3 x - 18}{x + 9}$ when $x = - 9$ .

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

$- 9$	$1$	$6$	$- 3$	$- 18$
	\xa0			\xa0

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

$- 9$	$1$	$6$	$- 3$	$- 18$
	\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 $(6)$ .

$- 9$	$1$	$6$	$- 3$	$- 18$
	\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.

$- 9$	$1$	$6$	$- 3$	$- 18$
	\xa0	$- 9$		\xa0
	$1$	$- 3$

Multiply the newest entry in the result $(- 3)$ by the divisor $(- 9)$ and place the result of $(27)$ under the next term in the dividend $(- 3)$ .

$- 9$	$1$	$6$	$- 3$	$- 18$
	\xa0	$- 9$	$27$	\xa0
	$1$	$- 3$

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

$- 9$	$1$	$6$	$- 3$	$- 18$
	\xa0	$- 9$	$27$	\xa0
	$1$	$- 3$	$24$

Multiply the newest entry in the result $(24)$ by the divisor $(- 9)$ and place the result of $(- 216)$ under the next term in the dividend $(- 18)$ .

$- 9$	$1$	$6$	$- 3$	$- 18$
	\xa0	$- 9$	$27$	$- 216$
	$1$	$- 3$	$24$

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

$- 9$	$1$	$6$	$- 3$	$- 18$
	\xa0	$- 9$	$27$	$- 216$
	$1$	$- 3$	$24$	$- 234$

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

$1 x^{2} + - 3 x + 24 + \frac{- 234}{x + 9}$

Simplify the quotient polynomial.

$x^{2} - 3 x + 24 - \frac{234}{x + 9}$

Step 13

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 14

Apply synthetic division on $\frac{x^{3} + 6 x^{2} - 3 x - 18}{x - 18}$ when $x = 18$ .

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

$18$	$1$	$6$	$- 3$	$- 18$
	\xa0			\xa0

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

$18$	$1$	$6$	$- 3$	$- 18$
	\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 $(6)$ .

$18$	$1$	$6$	$- 3$	$- 18$
	\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.

$18$	$1$	$6$	$- 3$	$- 18$
	\xa0	$18$		\xa0
	$1$	$24$

Multiply the newest entry in the result $(24)$ by the divisor $(18)$ and place the result of $(432)$ under the next term in the dividend $(- 3)$ .

$18$	$1$	$6$	$- 3$	$- 18$
	\xa0	$18$	$432$	\xa0
	$1$	$24$

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

$18$	$1$	$6$	$- 3$	$- 18$
	\xa0	$18$	$432$	\xa0
	$1$	$24$	$429$

Multiply the newest entry in the result $(429)$ by the divisor $(18)$ and place the result of $(7722)$ under the next term in the dividend $(- 18)$ .

$18$	$1$	$6$	$- 3$	$- 18$
	\xa0	$18$	$432$	$7722$
	$1$	$24$	$429$

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

$18$	$1$	$6$	$- 3$	$- 18$
	\xa0	$18$	$432$	$7722$
	$1$	$24$	$429$	$7704$

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

$1 x^{2} + 24 x + 429 + \frac{7704}{x - 18}$

Simplify the quotient polynomial.

$x^{2} + 24 x + 429 + \frac{7704}{x - 18}$

Step 15

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 16

Apply synthetic division on $\frac{x^{3} + 6 x^{2} - 3 x - 18}{x + 18}$ when $x = - 18$ .

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

$- 18$	$1$	$6$	$- 3$	$- 18$
	\xa0			\xa0

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

$- 18$	$1$	$6$	$- 3$	$- 18$
	\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 $(6)$ .

$- 18$	$1$	$6$	$- 3$	$- 18$
	\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.

$- 18$	$1$	$6$	$- 3$	$- 18$
	\xa0	$- 18$		\xa0
	$1$	$- 12$

Multiply the newest entry in the result $(- 12)$ by the divisor $(- 18)$ and place the result of $(216)$ under the next term in the dividend $(- 3)$ .

$- 18$	$1$	$6$	$- 3$	$- 18$
	\xa0	$- 18$	$216$	\xa0
	$1$	$- 12$

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

$- 18$	$1$	$6$	$- 3$	$- 18$
	\xa0	$- 18$	$216$	\xa0
	$1$	$- 12$	$213$

Multiply the newest entry in the result $(213)$ by the divisor $(- 18)$ and place the result of $(- 3834)$ under the next term in the dividend $(- 18)$ .

$- 18$	$1$	$6$	$- 3$	$- 18$
	\xa0	$- 18$	$216$	$- 3834$
	$1$	$- 12$	$213$

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

$- 18$	$1$	$6$	$- 3$	$- 18$
	\xa0	$- 18$	$216$	$- 3834$
	$1$	$- 12$	$213$	$- 3852$

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

$1 x^{2} + - 12 x + 213 + \frac{- 3852}{x + 18}$

Simplify the quotient polynomial.

$x^{2} - 12 x + 213 - \frac{3852}{x + 18}$

Step 17

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 18

Determine the upper and lower bounds.

Upper Bounds: $2, 3, 6, 9, 18$

Lower Bounds: $- 6, - 9, - 18$

Step 19

Find the Upper and Lower Bounds f(x)=x^3+6x^2-3x-18

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