Find the Upper and Lower Bounds f(x)=11x^4+11x^3-x^2-44x-44

$f (x) = 11 x^{4} + 11 x^{3} - x^{2} - 44 x - 44$

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.

$p = \pm 1, \pm 2, \pm 4, \pm 11, \pm 22, \pm 44$

$q = \pm 1, \pm 11$

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

$\pm 1, \pm \frac{1}{11}, \pm 2, \pm \frac{2}{11}, \pm 4, \pm \frac{4}{11}, \pm 11, \pm 22, \pm 44$

Step 2

Apply synthetic division on $\frac{11 x^{4} + 11 x^{3} - x^{2} - 44 x - 44}{x - 2}$ when $x = 2$ .

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

$2$	$11$	$11$	$- 1$	$- 44$	$- 44$
	\xa0				\xa0

The first number in the dividend $(11)$ is put into the first position of the result area (below the horizontal line).

$2$	$11$	$11$	$- 1$	$- 44$	$- 44$
	\xa0				\xa0
	$11$

Multiply the newest entry in the result $(11)$ by the divisor $(2)$ and place the result of $(22)$ under the next term in the dividend $(11)$ .

$2$	$11$	$11$	$- 1$	$- 44$	$- 44$
	\xa0	$22$			\xa0
	$11$

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

$2$	$11$	$11$	$- 1$	$- 44$	$- 44$
	\xa0	$22$			\xa0
	$11$	$33$

Multiply the newest entry in the result $(33)$ by the divisor $(2)$ and place the result of $(66)$ under the next term in the dividend $(- 1)$ .

$2$	$11$	$11$	$- 1$	$- 44$	$- 44$
	\xa0	$22$	$66$		\xa0
	$11$	$33$

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

$2$	$11$	$11$	$- 1$	$- 44$	$- 44$
	\xa0	$22$	$66$		\xa0
	$11$	$33$	$65$

Multiply the newest entry in the result $(65)$ by the divisor $(2)$ and place the result of $(130)$ under the next term in the dividend $(- 44)$ .

$2$	$11$	$11$	$- 1$	$- 44$	$- 44$
	\xa0	$22$	$66$	$130$	\xa0
	$11$	$33$	$65$

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

$2$	$11$	$11$	$- 1$	$- 44$	$- 44$
	\xa0	$22$	$66$	$130$	\xa0
	$11$	$33$	$65$	$86$

Multiply the newest entry in the result $(86)$ by the divisor $(2)$ and place the result of $(172)$ under the next term in the dividend $(- 44)$ .

$2$	$11$	$11$	$- 1$	$- 44$	$- 44$
	\xa0	$22$	$66$	$130$	$172$
	$11$	$33$	$65$	$86$

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

$2$	$11$	$11$	$- 1$	$- 44$	$- 44$
	\xa0	$22$	$66$	$130$	$172$
	$11$	$33$	$65$	$86$	$128$

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

$11 x^{3} + 33 x^{2} + (65) x + 86 + \frac{128}{x - 2}$

Simplify the quotient polynomial.

$11 x^{3} + 33 x^{2} + 65 x + 86 + \frac{128}{x - 2}$

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 $\frac{11 x^{4} + 11 x^{3} - x^{2} - 44 x - 44}{x + 2}$ when $x = - 2$ .

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

$- 2$	$11$	$11$	$- 1$	$- 44$	$- 44$
	\xa0				\xa0

The first number in the dividend $(11)$ is put into the first position of the result area (below the horizontal line).

$- 2$	$11$	$11$	$- 1$	$- 44$	$- 44$
	\xa0				\xa0
	$11$

Multiply the newest entry in the result $(11)$ by the divisor $(- 2)$ and place the result of $(- 22)$ under the next term in the dividend $(11)$ .

$- 2$	$11$	$11$	$- 1$	$- 44$	$- 44$
	\xa0	$- 22$			\xa0
	$11$

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

$- 2$	$11$	$11$	$- 1$	$- 44$	$- 44$
	\xa0	$- 22$			\xa0
	$11$	$- 11$

Multiply the newest entry in the result $(- 11)$ by the divisor $(- 2)$ and place the result of $(22)$ under the next term in the dividend $(- 1)$ .

$- 2$	$11$	$11$	$- 1$	$- 44$	$- 44$
	\xa0	$- 22$	$22$		\xa0
	$11$	$- 11$

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

$- 2$	$11$	$11$	$- 1$	$- 44$	$- 44$
	\xa0	$- 22$	$22$		\xa0
	$11$	$- 11$	$21$

Multiply the newest entry in the result $(21)$ by the divisor $(- 2)$ and place the result of $(- 42)$ under the next term in the dividend $(- 44)$ .

$- 2$	$11$	$11$	$- 1$	$- 44$	$- 44$
	\xa0	$- 22$	$22$	$- 42$	\xa0
	$11$	$- 11$	$21$

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

$- 2$	$11$	$11$	$- 1$	$- 44$	$- 44$
	\xa0	$- 22$	$22$	$- 42$	\xa0
	$11$	$- 11$	$21$	$- 86$

Multiply the newest entry in the result $(- 86)$ by the divisor $(- 2)$ and place the result of $(172)$ under the next term in the dividend $(- 44)$ .

$- 2$	$11$	$11$	$- 1$	$- 44$	$- 44$
	\xa0	$- 22$	$22$	$- 42$	$172$
	$11$	$- 11$	$21$	$- 86$

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

$- 2$	$11$	$11$	$- 1$	$- 44$	$- 44$
	\xa0	$- 22$	$22$	$- 42$	$172$
	$11$	$- 11$	$21$	$- 86$	$128$

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

$11 x^{3} + - 11 x^{2} + (21) x - 86 + \frac{128}{x + 2}$

Simplify the quotient polynomial.

$11 x^{3} - 11 x^{2} + 21 x - 86 + \frac{128}{x + 2}$

Step 5

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 6

Apply synthetic division on $\frac{11 x^{4} + 11 x^{3} - x^{2} - 44 x - 44}{x - 4}$ when $x = 4$ .

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

$4$	$11$	$11$	$- 1$	$- 44$	$- 44$
	\xa0				\xa0

The first number in the dividend $(11)$ is put into the first position of the result area (below the horizontal line).

$4$	$11$	$11$	$- 1$	$- 44$	$- 44$
	\xa0				\xa0
	$11$

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 $(11)$ .

$4$	$11$	$11$	$- 1$	$- 44$	$- 44$
	\xa0	$44$			\xa0
	$11$

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

$4$	$11$	$11$	$- 1$	$- 44$	$- 44$
	\xa0	$44$			\xa0
	$11$	$55$

Multiply the newest entry in the result $(55)$ by the divisor $(4)$ and place the result of $(220)$ under the next term in the dividend $(- 1)$ .

$4$	$11$	$11$	$- 1$	$- 44$	$- 44$
	\xa0	$44$	$220$		\xa0
	$11$	$55$

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

$4$	$11$	$11$	$- 1$	$- 44$	$- 44$
	\xa0	$44$	$220$		\xa0
	$11$	$55$	$219$

Multiply the newest entry in the result $(219)$ by the divisor $(4)$ and place the result of $(876)$ under the next term in the dividend $(- 44)$ .

$4$	$11$	$11$	$- 1$	$- 44$	$- 44$
	\xa0	$44$	$220$	$876$	\xa0
	$11$	$55$	$219$

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

$4$	$11$	$11$	$- 1$	$- 44$	$- 44$
	\xa0	$44$	$220$	$876$	\xa0
	$11$	$55$	$219$	$832$

Multiply the newest entry in the result $(832)$ by the divisor $(4)$ and place the result of $(3328)$ under the next term in the dividend $(- 44)$ .

$4$	$11$	$11$	$- 1$	$- 44$	$- 44$
	\xa0	$44$	$220$	$876$	$3328$
	$11$	$55$	$219$	$832$

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

$4$	$11$	$11$	$- 1$	$- 44$	$- 44$
	\xa0	$44$	$220$	$876$	$3328$
	$11$	$55$	$219$	$832$	$3284$

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

$11 x^{3} + 55 x^{2} + (219) x + 832 + \frac{3284}{x - 4}$

Simplify the quotient polynomial.

$11 x^{3} + 55 x^{2} + 219 x + 832 + \frac{3284}{x - 4}$

Step 7

Since $4 > 0$ and all of the signs in the bottom row of the synthetic division are positive, $4$ is an upper bound for the real roots of the function.

Upper Bound: $4$

Step 8

Apply synthetic division on $\frac{11 x^{4} + 11 x^{3} - x^{2} - 44 x - 44}{x + 4}$ when $x = - 4$ .

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

$- 4$	$11$	$11$	$- 1$	$- 44$	$- 44$
	\xa0				\xa0

The first number in the dividend $(11)$ is put into the first position of the result area (below the horizontal line).

$- 4$	$11$	$11$	$- 1$	$- 44$	$- 44$
	\xa0				\xa0
	$11$

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 $(11)$ .

$- 4$	$11$	$11$	$- 1$	$- 44$	$- 44$
	\xa0	$- 44$			\xa0
	$11$

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

$- 4$	$11$	$11$	$- 1$	$- 44$	$- 44$
	\xa0	$- 44$			\xa0
	$11$	$- 33$

Multiply the newest entry in the result $(- 33)$ by the divisor $(- 4)$ and place the result of $(132)$ under the next term in the dividend $(- 1)$ .

$- 4$	$11$	$11$	$- 1$	$- 44$	$- 44$
	\xa0	$- 44$	$132$		\xa0
	$11$	$- 33$

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

$- 4$	$11$	$11$	$- 1$	$- 44$	$- 44$
	\xa0	$- 44$	$132$		\xa0
	$11$	$- 33$	$131$

Multiply the newest entry in the result $(131)$ by the divisor $(- 4)$ and place the result of $(- 524)$ under the next term in the dividend $(- 44)$ .

$- 4$	$11$	$11$	$- 1$	$- 44$	$- 44$
	\xa0	$- 44$	$132$	$- 524$	\xa0
	$11$	$- 33$	$131$

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

$- 4$	$11$	$11$	$- 1$	$- 44$	$- 44$
	\xa0	$- 44$	$132$	$- 524$	\xa0
	$11$	$- 33$	$131$	$- 568$

Multiply the newest entry in the result $(- 568)$ by the divisor $(- 4)$ and place the result of $(2272)$ under the next term in the dividend $(- 44)$ .

$- 4$	$11$	$11$	$- 1$	$- 44$	$- 44$
	\xa0	$- 44$	$132$	$- 524$	$2272$
	$11$	$- 33$	$131$	$- 568$

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

$- 4$	$11$	$11$	$- 1$	$- 44$	$- 44$
	\xa0	$- 44$	$132$	$- 524$	$2272$
	$11$	$- 33$	$131$	$- 568$	$2228$

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

$11 x^{3} + - 33 x^{2} + (131) x - 568 + \frac{2228}{x + 4}$

Simplify the quotient polynomial.

$11 x^{3} - 33 x^{2} + 131 x - 568 + \frac{2228}{x + 4}$

Step 9

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 10

Apply synthetic division on $\frac{11 x^{4} + 11 x^{3} - x^{2} - 44 x - 44}{x - 11}$ when $x = 11$ .

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

$11$	$11$	$11$	$- 1$	$- 44$	$- 44$
	\xa0				\xa0

The first number in the dividend $(11)$ is put into the first position of the result area (below the horizontal line).

$11$	$11$	$11$	$- 1$	$- 44$	$- 44$
	\xa0				\xa0
	$11$

Multiply the newest entry in the result $(11)$ by the divisor $(11)$ and place the result of $(121)$ under the next term in the dividend $(11)$ .

$11$	$11$	$11$	$- 1$	$- 44$	$- 44$
	\xa0	$121$			\xa0
	$11$

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

$11$	$11$	$11$	$- 1$	$- 44$	$- 44$
	\xa0	$121$			\xa0
	$11$	$132$

Multiply the newest entry in the result $(132)$ by the divisor $(11)$ and place the result of $(1452)$ under the next term in the dividend $(- 1)$ .

$11$	$11$	$11$	$- 1$	$- 44$	$- 44$
	\xa0	$121$	$1452$		\xa0
	$11$	$132$

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

$11$	$11$	$11$	$- 1$	$- 44$	$- 44$
	\xa0	$121$	$1452$		\xa0
	$11$	$132$	$1451$

Multiply the newest entry in the result $(1451)$ by the divisor $(11)$ and place the result of $(15961)$ under the next term in the dividend $(- 44)$ .

$11$	$11$	$11$	$- 1$	$- 44$	$- 44$
	\xa0	$121$	$1452$	$15961$	\xa0
	$11$	$132$	$1451$

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

$11$	$11$	$11$	$- 1$	$- 44$	$- 44$
	\xa0	$121$	$1452$	$15961$	\xa0
	$11$	$132$	$1451$	$15917$

Multiply the newest entry in the result $(15917)$ by the divisor $(11)$ and place the result of $(175087)$ under the next term in the dividend $(- 44)$ .

$11$	$11$	$11$	$- 1$	$- 44$	$- 44$
	\xa0	$121$	$1452$	$15961$	$175087$
	$11$	$132$	$1451$	$15917$

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

$11$	$11$	$11$	$- 1$	$- 44$	$- 44$
	\xa0	$121$	$1452$	$15961$	$175087$
	$11$	$132$	$1451$	$15917$	$175043$

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

$11 x^{3} + 132 x^{2} + (1451) x + 15917 + \frac{175043}{x - 11}$

Simplify the quotient polynomial.

$11 x^{3} + 132 x^{2} + 1451 x + 15917 + \frac{175043}{x - 11}$

Step 11

Since $11 > 0$ and all of the signs in the bottom row of the synthetic division are positive, $11$ is an upper bound for the real roots of the function.

Upper Bound: $11$

Step 12

Apply synthetic division on $\frac{11 x^{4} + 11 x^{3} - x^{2} - 44 x - 44}{x + 11}$ when $x = - 11$ .

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

$- 11$	$11$	$11$	$- 1$	$- 44$	$- 44$
	\xa0				\xa0

The first number in the dividend $(11)$ is put into the first position of the result area (below the horizontal line).

$- 11$	$11$	$11$	$- 1$	$- 44$	$- 44$
	\xa0				\xa0
	$11$

Multiply the newest entry in the result $(11)$ by the divisor $(- 11)$ and place the result of $(- 121)$ under the next term in the dividend $(11)$ .

$- 11$	$11$	$11$	$- 1$	$- 44$	$- 44$
	\xa0	$- 121$			\xa0
	$11$

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

$- 11$	$11$	$11$	$- 1$	$- 44$	$- 44$
	\xa0	$- 121$			\xa0
	$11$	$- 110$

Multiply the newest entry in the result $(- 110)$ by the divisor $(- 11)$ and place the result of $(1210)$ under the next term in the dividend $(- 1)$ .

$- 11$	$11$	$11$	$- 1$	$- 44$	$- 44$
	\xa0	$- 121$	$1210$		\xa0
	$11$	$- 110$

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

$- 11$	$11$	$11$	$- 1$	$- 44$	$- 44$
	\xa0	$- 121$	$1210$		\xa0
	$11$	$- 110$	$1209$

Multiply the newest entry in the result $(1209)$ by the divisor $(- 11)$ and place the result of $(- 13299)$ under the next term in the dividend $(- 44)$ .

$- 11$	$11$	$11$	$- 1$	$- 44$	$- 44$
	\xa0	$- 121$	$1210$	$- 13299$	\xa0
	$11$	$- 110$	$1209$

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

$- 11$	$11$	$11$	$- 1$	$- 44$	$- 44$
	\xa0	$- 121$	$1210$	$- 13299$	\xa0
	$11$	$- 110$	$1209$	$- 13343$

Multiply the newest entry in the result $(- 13343)$ by the divisor $(- 11)$ and place the result of $(146773)$ under the next term in the dividend $(- 44)$ .

$- 11$	$11$	$11$	$- 1$	$- 44$	$- 44$
	\xa0	$- 121$	$1210$	$- 13299$	$146773$
	$11$	$- 110$	$1209$	$- 13343$

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

$- 11$	$11$	$11$	$- 1$	$- 44$	$- 44$
	\xa0	$- 121$	$1210$	$- 13299$	$146773$
	$11$	$- 110$	$1209$	$- 13343$	$146729$

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

$11 x^{3} + - 110 x^{2} + (1209) x - 13343 + \frac{146729}{x + 11}$

Simplify the quotient polynomial.

$11 x^{3} - 110 x^{2} + 1209 x - 13343 + \frac{146729}{x + 11}$

Step 13

Since $- 11 < 0$ and the signs in the bottom row of the synthetic division alternate sign, $- 11$ is a lower bound for the real roots of the function.

Lower Bound: $- 11$

Step 14

Apply synthetic division on $\frac{11 x^{4} + 11 x^{3} - x^{2} - 44 x - 44}{x - 22}$ when $x = 22$ .

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

$22$	$11$	$11$	$- 1$	$- 44$	$- 44$
	\xa0				\xa0

The first number in the dividend $(11)$ is put into the first position of the result area (below the horizontal line).

$22$	$11$	$11$	$- 1$	$- 44$	$- 44$
	\xa0				\xa0
	$11$

Multiply the newest entry in the result $(11)$ by the divisor $(22)$ and place the result of $(242)$ under the next term in the dividend $(11)$ .

$22$	$11$	$11$	$- 1$	$- 44$	$- 44$
	\xa0	$242$			\xa0
	$11$

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

$22$	$11$	$11$	$- 1$	$- 44$	$- 44$
	\xa0	$242$			\xa0
	$11$	$253$

Multiply the newest entry in the result $(253)$ by the divisor $(22)$ and place the result of $(5566)$ under the next term in the dividend $(- 1)$ .

$22$	$11$	$11$	$- 1$	$- 44$	$- 44$
	\xa0	$242$	$5566$		\xa0
	$11$	$253$

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

$22$	$11$	$11$	$- 1$	$- 44$	$- 44$
	\xa0	$242$	$5566$		\xa0
	$11$	$253$	$5565$

Multiply the newest entry in the result $(5565)$ by the divisor $(22)$ and place the result of $(122430)$ under the next term in the dividend $(- 44)$ .

$22$	$11$	$11$	$- 1$	$- 44$	$- 44$
	\xa0	$242$	$5566$	$122430$	\xa0
	$11$	$253$	$5565$

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

$22$	$11$	$11$	$- 1$	$- 44$	$- 44$
	\xa0	$242$	$5566$	$122430$	\xa0
	$11$	$253$	$5565$	$122386$

Multiply the newest entry in the result $(122386)$ by the divisor $(22)$ and place the result of $(2692492)$ under the next term in the dividend $(- 44)$ .

$22$	$11$	$11$	$- 1$	$- 44$	$- 44$
	\xa0	$242$	$5566$	$122430$	$2692492$
	$11$	$253$	$5565$	$122386$

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

$22$	$11$	$11$	$- 1$	$- 44$	$- 44$
	\xa0	$242$	$5566$	$122430$	$2692492$
	$11$	$253$	$5565$	$122386$	$2692448$

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

$11 x^{3} + 253 x^{2} + (5565) x + 122386 + \frac{2692448}{x - 22}$

Simplify the quotient polynomial.

$11 x^{3} + 253 x^{2} + 5565 x + 122386 + \frac{2692448}{x - 22}$

Step 15

Since $22 > 0$ and all of the signs in the bottom row of the synthetic division are positive, $22$ is an upper bound for the real roots of the function.

Upper Bound: $22$

Step 16

Apply synthetic division on $\frac{11 x^{4} + 11 x^{3} - x^{2} - 44 x - 44}{x + 22}$ when $x = - 22$ .

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

$- 22$	$11$	$11$	$- 1$	$- 44$	$- 44$
	\xa0				\xa0

The first number in the dividend $(11)$ is put into the first position of the result area (below the horizontal line).

$- 22$	$11$	$11$	$- 1$	$- 44$	$- 44$
	\xa0				\xa0
	$11$

Multiply the newest entry in the result $(11)$ by the divisor $(- 22)$ and place the result of $(- 242)$ under the next term in the dividend $(11)$ .

$- 22$	$11$	$11$	$- 1$	$- 44$	$- 44$
	\xa0	$- 242$			\xa0
	$11$

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

$- 22$	$11$	$11$	$- 1$	$- 44$	$- 44$
	\xa0	$- 242$			\xa0
	$11$	$- 231$

Multiply the newest entry in the result $(- 231)$ by the divisor $(- 22)$ and place the result of $(5082)$ under the next term in the dividend $(- 1)$ .

$- 22$	$11$	$11$	$- 1$	$- 44$	$- 44$
	\xa0	$- 242$	$5082$		\xa0
	$11$	$- 231$

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

$- 22$	$11$	$11$	$- 1$	$- 44$	$- 44$
	\xa0	$- 242$	$5082$		\xa0
	$11$	$- 231$	$5081$

Multiply the newest entry in the result $(5081)$ by the divisor $(- 22)$ and place the result of $(- 111782)$ under the next term in the dividend $(- 44)$ .

$- 22$	$11$	$11$	$- 1$	$- 44$	$- 44$
	\xa0	$- 242$	$5082$	$- 111782$	\xa0
	$11$	$- 231$	$5081$

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

$- 22$	$11$	$11$	$- 1$	$- 44$	$- 44$
	\xa0	$- 242$	$5082$	$- 111782$	\xa0
	$11$	$- 231$	$5081$	$- 111826$

Multiply the newest entry in the result $(- 111826)$ by the divisor $(- 22)$ and place the result of $(2460172)$ under the next term in the dividend $(- 44)$ .

$- 22$	$11$	$11$	$- 1$	$- 44$	$- 44$
	\xa0	$- 242$	$5082$	$- 111782$	$2460172$
	$11$	$- 231$	$5081$	$- 111826$

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

$- 22$	$11$	$11$	$- 1$	$- 44$	$- 44$
	\xa0	$- 242$	$5082$	$- 111782$	$2460172$
	$11$	$- 231$	$5081$	$- 111826$	$2460128$

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

$11 x^{3} + - 231 x^{2} + (5081) x - 111826 + \frac{2460128}{x + 22}$

Simplify the quotient polynomial.

$11 x^{3} - 231 x^{2} + 5081 x - 111826 + \frac{2460128}{x + 22}$

Step 17

Since $- 22 < 0$ and the signs in the bottom row of the synthetic division alternate sign, $- 22$ is a lower bound for the real roots of the function.

Lower Bound: $- 22$

Step 18

Apply synthetic division on $\frac{11 x^{4} + 11 x^{3} - x^{2} - 44 x - 44}{x - 44}$ when $x = 44$ .

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

$44$	$11$	$11$	$- 1$	$- 44$	$- 44$
	\xa0				\xa0

The first number in the dividend $(11)$ is put into the first position of the result area (below the horizontal line).

$44$	$11$	$11$	$- 1$	$- 44$	$- 44$
	\xa0				\xa0
	$11$

Multiply the newest entry in the result $(11)$ by the divisor $(44)$ and place the result of $(484)$ under the next term in the dividend $(11)$ .

$44$	$11$	$11$	$- 1$	$- 44$	$- 44$
	\xa0	$484$			\xa0
	$11$

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

$44$	$11$	$11$	$- 1$	$- 44$	$- 44$
	\xa0	$484$			\xa0
	$11$	$495$

Multiply the newest entry in the result $(495)$ by the divisor $(44)$ and place the result of $(21780)$ under the next term in the dividend $(- 1)$ .

$44$	$11$	$11$	$- 1$	$- 44$	$- 44$
	\xa0	$484$	$21780$		\xa0
	$11$	$495$

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

$44$	$11$	$11$	$- 1$	$- 44$	$- 44$
	\xa0	$484$	$21780$		\xa0
	$11$	$495$	$21779$

Multiply the newest entry in the result $(21779)$ by the divisor $(44)$ and place the result of $(958276)$ under the next term in the dividend $(- 44)$ .

$44$	$11$	$11$	$- 1$	$- 44$	$- 44$
	\xa0	$484$	$21780$	$958276$	\xa0
	$11$	$495$	$21779$

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

$44$	$11$	$11$	$- 1$	$- 44$	$- 44$
	\xa0	$484$	$21780$	$958276$	\xa0
	$11$	$495$	$21779$	$958232$

Multiply the newest entry in the result $(958232)$ by the divisor $(44)$ and place the result of $(42162208)$ under the next term in the dividend $(- 44)$ .

$44$	$11$	$11$	$- 1$	$- 44$	$- 44$
	\xa0	$484$	$21780$	$958276$	$42162208$
	$11$	$495$	$21779$	$958232$

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

$44$	$11$	$11$	$- 1$	$- 44$	$- 44$
	\xa0	$484$	$21780$	$958276$	$42162208$
	$11$	$495$	$21779$	$958232$	$42162164$

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

$11 x^{3} + 495 x^{2} + (21779) x + 958232 + \frac{42162164}{x - 44}$

Simplify the quotient polynomial.

$11 x^{3} + 495 x^{2} + 21779 x + 958232 + \frac{42162164}{x - 44}$

Step 19

Since $44 > 0$ and all of the signs in the bottom row of the synthetic division are positive, $44$ is an upper bound for the real roots of the function.

Upper Bound: $44$

Step 20

Apply synthetic division on $\frac{11 x^{4} + 11 x^{3} - x^{2} - 44 x - 44}{x + 44}$ when $x = - 44$ .

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

$- 44$	$11$	$11$	$- 1$	$- 44$	$- 44$
	\xa0				\xa0

The first number in the dividend $(11)$ is put into the first position of the result area (below the horizontal line).

$- 44$	$11$	$11$	$- 1$	$- 44$	$- 44$
	\xa0				\xa0
	$11$

Multiply the newest entry in the result $(11)$ by the divisor $(- 44)$ and place the result of $(- 484)$ under the next term in the dividend $(11)$ .

$- 44$	$11$	$11$	$- 1$	$- 44$	$- 44$
	\xa0	$- 484$			\xa0
	$11$

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

$- 44$	$11$	$11$	$- 1$	$- 44$	$- 44$
	\xa0	$- 484$			\xa0
	$11$	$- 473$

Multiply the newest entry in the result $(- 473)$ by the divisor $(- 44)$ and place the result of $(20812)$ under the next term in the dividend $(- 1)$ .

$- 44$	$11$	$11$	$- 1$	$- 44$	$- 44$
	\xa0	$- 484$	$20812$		\xa0
	$11$	$- 473$

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

$- 44$	$11$	$11$	$- 1$	$- 44$	$- 44$
	\xa0	$- 484$	$20812$		\xa0
	$11$	$- 473$	$20811$

Multiply the newest entry in the result $(20811)$ by the divisor $(- 44)$ and place the result of $(- 915684)$ under the next term in the dividend $(- 44)$ .

$- 44$	$11$	$11$	$- 1$	$- 44$	$- 44$
	\xa0	$- 484$	$20812$	$- 915684$	\xa0
	$11$	$- 473$	$20811$

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

$- 44$	$11$	$11$	$- 1$	$- 44$	$- 44$
	\xa0	$- 484$	$20812$	$- 915684$	\xa0
	$11$	$- 473$	$20811$	$- 915728$

Multiply the newest entry in the result $(- 915728)$ by the divisor $(- 44)$ and place the result of $(40292032)$ under the next term in the dividend $(- 44)$ .

$- 44$	$11$	$11$	$- 1$	$- 44$	$- 44$
	\xa0	$- 484$	$20812$	$- 915684$	$40292032$
	$11$	$- 473$	$20811$	$- 915728$

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

$- 44$	$11$	$11$	$- 1$	$- 44$	$- 44$
	\xa0	$- 484$	$20812$	$- 915684$	$40292032$
	$11$	$- 473$	$20811$	$- 915728$	$40291988$

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

$11 x^{3} + - 473 x^{2} + (20811) x - 915728 + \frac{40291988}{x + 44}$

Simplify the quotient polynomial.

$11 x^{3} - 473 x^{2} + 20811 x - 915728 + \frac{40291988}{x + 44}$

Step 21

Since $- 44 < 0$ and the signs in the bottom row of the synthetic division alternate sign, $- 44$ is a lower bound for the real roots of the function.

Lower Bound: $- 44$

Step 22

Determine the upper and lower bounds.

Upper Bounds: $2, 4, 11, 22, 44$

Lower Bounds: $- 2, - 4, - 11, - 22, - 44$

Step 23

Find the Upper and Lower Bounds f(x)=11x^4+11x^3-x^2-44x-44

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