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

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

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.

$p = \pm 1, \pm 11$

$q = \pm 1$

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

$\pm 1, \pm 11$

Step 2

Apply synthetic division on $\frac{x^{4} + x^{3} - 11 x - 11}{x + 1}$ when $x = - 1$ .

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

$- 1$	$1$	$1$	$0$	$- 11$	$- 11$
	\xa0				\xa0

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

$- 1$	$1$	$1$	$0$	$- 11$	$- 11$
	\xa0				\xa0
	$1$

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

$- 1$	$1$	$1$	$0$	$- 11$	$- 11$
	\xa0	$- 1$			\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.

$- 1$	$1$	$1$	$0$	$- 11$	$- 11$
	\xa0	$- 1$			\xa0
	$1$	$0$

Multiply the newest entry in the result $(0)$ by the divisor $(- 1)$ and place the result of $(0)$ under the next term in the dividend $(0)$ .

$- 1$	$1$	$1$	$0$	$- 11$	$- 11$
	\xa0	$- 1$	$0$		\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.

$- 1$	$1$	$1$	$0$	$- 11$	$- 11$
	\xa0	$- 1$	$0$		\xa0
	$1$	$0$	$0$

Multiply the newest entry in the result $(0)$ by the divisor $(- 1)$ and place the result of $(0)$ under the next term in the dividend $(- 11)$ .

$- 1$	$1$	$1$	$0$	$- 11$	$- 11$
	\xa0	$- 1$	$0$	$0$	\xa0
	$1$	$0$	$0$

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

$- 1$	$1$	$1$	$0$	$- 11$	$- 11$
	\xa0	$- 1$	$0$	$0$	\xa0
	$1$	$0$	$0$	$- 11$

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

$- 1$	$1$	$1$	$0$	$- 11$	$- 11$
	\xa0	$- 1$	$0$	$0$	$11$
	$1$	$0$	$0$	$- 11$

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

$- 1$	$1$	$1$	$0$	$- 11$	$- 11$
	\xa0	$- 1$	$0$	$0$	$11$
	$1$	$0$	$0$	$- 11$	$0$

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

$1 x^{3} + 0 x^{2} + (0) x - 11$

Simplify the quotient polynomial.

$x^{3} - 11$

Step 3

Since $- 1 < 0$ and the signs in the bottom row of the synthetic division alternate sign, $- 1$ is a lower bound for the real roots of the function.

Lower Bound: $- 1$

Step 4

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

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

$11$	$1$	$1$	$0$	$- 11$	$- 11$
	\xa0				\xa0

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

$11$	$1$	$1$	$0$	$- 11$	$- 11$
	\xa0				\xa0
	$1$

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

$11$	$1$	$1$	$0$	$- 11$	$- 11$
	\xa0	$11$			\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.

$11$	$1$	$1$	$0$	$- 11$	$- 11$
	\xa0	$11$			\xa0
	$1$	$12$

Multiply the newest entry in the result $(12)$ by the divisor $(11)$ and place the result of $(132)$ under the next term in the dividend $(0)$ .

$11$	$1$	$1$	$0$	$- 11$	$- 11$
	\xa0	$11$	$132$		\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.

$11$	$1$	$1$	$0$	$- 11$	$- 11$
	\xa0	$11$	$132$		\xa0
	$1$	$12$	$132$

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 $(- 11)$ .

$11$	$1$	$1$	$0$	$- 11$	$- 11$
	\xa0	$11$	$132$	$1452$	\xa0
	$1$	$12$	$132$

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

$11$	$1$	$1$	$0$	$- 11$	$- 11$
	\xa0	$11$	$132$	$1452$	\xa0
	$1$	$12$	$132$	$1441$

Multiply the newest entry in the result $(1441)$ by the divisor $(11)$ and place the result of $(15851)$ under the next term in the dividend $(- 11)$ .

$11$	$1$	$1$	$0$	$- 11$	$- 11$
	\xa0	$11$	$132$	$1452$	$15851$
	$1$	$12$	$132$	$1441$

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

$11$	$1$	$1$	$0$	$- 11$	$- 11$
	\xa0	$11$	$132$	$1452$	$15851$
	$1$	$12$	$132$	$1441$	$15840$

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

$1 x^{3} + 12 x^{2} + (132) x + 1441 + \frac{15840}{x - 11}$

Simplify the quotient polynomial.

$x^{3} + 12 x^{2} + 132 x + 1441 + \frac{15840}{x - 11}$

Step 5

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 6

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

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

$- 11$	$1$	$1$	$0$	$- 11$	$- 11$
	\xa0				\xa0

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

$- 11$	$1$	$1$	$0$	$- 11$	$- 11$
	\xa0				\xa0
	$1$

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

$- 11$	$1$	$1$	$0$	$- 11$	$- 11$
	\xa0	$- 11$			\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.

$- 11$	$1$	$1$	$0$	$- 11$	$- 11$
	\xa0	$- 11$			\xa0
	$1$	$- 10$

Multiply the newest entry in the result $(- 10)$ by the divisor $(- 11)$ and place the result of $(110)$ under the next term in the dividend $(0)$ .

$- 11$	$1$	$1$	$0$	$- 11$	$- 11$
	\xa0	$- 11$	$110$		\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.

$- 11$	$1$	$1$	$0$	$- 11$	$- 11$
	\xa0	$- 11$	$110$		\xa0
	$1$	$- 10$	$110$

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 $(- 11)$ .

$- 11$	$1$	$1$	$0$	$- 11$	$- 11$
	\xa0	$- 11$	$110$	$- 1210$	\xa0
	$1$	$- 10$	$110$

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

$- 11$	$1$	$1$	$0$	$- 11$	$- 11$
	\xa0	$- 11$	$110$	$- 1210$	\xa0
	$1$	$- 10$	$110$	$- 1221$

Multiply the newest entry in the result $(- 1221)$ by the divisor $(- 11)$ and place the result of $(13431)$ under the next term in the dividend $(- 11)$ .

$- 11$	$1$	$1$	$0$	$- 11$	$- 11$
	\xa0	$- 11$	$110$	$- 1210$	$13431$
	$1$	$- 10$	$110$	$- 1221$

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

$- 11$	$1$	$1$	$0$	$- 11$	$- 11$
	\xa0	$- 11$	$110$	$- 1210$	$13431$
	$1$	$- 10$	$110$	$- 1221$	$13420$

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

$1 x^{3} + - 10 x^{2} + (110) x - 1221 + \frac{13420}{x + 11}$

Simplify the quotient polynomial.

$x^{3} - 10 x^{2} + 110 x - 1221 + \frac{13420}{x + 11}$

Step 7

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 8

Determine the upper and lower bounds.

Upper Bound: $11$

Lower Bounds: $- 1, - 11$

Step 9

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

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