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

$x^{4} + x^{3} - 2 x - 2$

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.

$p = \pm 1, \pm 2$

$q = \pm 1$

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

$\pm 1, \pm 2$

Step 2

Apply synthetic division on $\frac{x^{4} + x^{3} - 2 x - 2}{x + 1}$ when $x = - 1$ .

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

$- 1$	$1$	$1$	$0$	$- 2$	$- 2$
	\xa0				\xa0

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

$- 1$	$1$	$1$	$0$	$- 2$	$- 2$
	\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 $(1)$ .

$- 1$	$1$	$1$	$0$	$- 2$	$- 2$
	\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.

$- 1$	$1$	$1$	$0$	$- 2$	$- 2$
	\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 $(0)$ .

$- 1$	$1$	$1$	$0$	$- 2$	$- 2$
	\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.

$- 1$	$1$	$1$	$0$	$- 2$	$- 2$
	\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 $(- 2)$ .

$- 1$	$1$	$1$	$0$	$- 2$	$- 2$
	\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.

$- 1$	$1$	$1$	$0$	$- 2$	$- 2$
	\xa0	$- 1$	$0$	$0$	\xa0
	$1$	$0$	$0$	$- 2$

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

$- 1$	$1$	$1$	$0$	$- 2$	$- 2$
	\xa0	$- 1$	$0$	$0$	$2$
	$1$	$0$	$0$	$- 2$

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

$- 1$	$1$	$1$	$0$	$- 2$	$- 2$
	\xa0	$- 1$	$0$	$0$	$2$
	$1$	$0$	$0$	$- 2$	$0$

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

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

Simplify the quotient polynomial.

$x^{3} - 2$

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

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

$2$	$1$	$1$	$0$	$- 2$	$- 2$
	\xa0				\xa0

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

$2$	$1$	$1$	$0$	$- 2$	$- 2$
	\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 $(1)$ .

$2$	$1$	$1$	$0$	$- 2$	$- 2$
	\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.

$2$	$1$	$1$	$0$	$- 2$	$- 2$
	\xa0	$2$			\xa0
	$1$	$3$

Multiply the newest entry in the result $(3)$ by the divisor $(2)$ and place the result of $(6)$ under the next term in the dividend $(0)$ .

$2$	$1$	$1$	$0$	$- 2$	$- 2$
	\xa0	$2$	$6$		\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.

$2$	$1$	$1$	$0$	$- 2$	$- 2$
	\xa0	$2$	$6$		\xa0
	$1$	$3$	$6$

Multiply the newest entry in the result $(6)$ by the divisor $(2)$ and place the result of $(12)$ under the next term in the dividend $(- 2)$ .

$2$	$1$	$1$	$0$	$- 2$	$- 2$
	\xa0	$2$	$6$	$12$	\xa0
	$1$	$3$	$6$

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

$2$	$1$	$1$	$0$	$- 2$	$- 2$
	\xa0	$2$	$6$	$12$	\xa0
	$1$	$3$	$6$	$10$

Multiply the newest entry in the result $(10)$ by the divisor $(2)$ and place the result of $(20)$ under the next term in the dividend $(- 2)$ .

$2$	$1$	$1$	$0$	$- 2$	$- 2$
	\xa0	$2$	$6$	$12$	$20$
	$1$	$3$	$6$	$10$

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

$2$	$1$	$1$	$0$	$- 2$	$- 2$
	\xa0	$2$	$6$	$12$	$20$
	$1$	$3$	$6$	$10$	$18$

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

$1 x^{3} + 3 x^{2} + (6) x + 10 + \frac{18}{x - 2}$

Simplify the quotient polynomial.

$x^{3} + 3 x^{2} + 6 x + 10 + \frac{18}{x - 2}$

Step 5

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 6

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

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

$- 2$	$1$	$1$	$0$	$- 2$	$- 2$
	\xa0				\xa0

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

$- 2$	$1$	$1$	$0$	$- 2$	$- 2$
	\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 $(1)$ .

$- 2$	$1$	$1$	$0$	$- 2$	$- 2$
	\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.

$- 2$	$1$	$1$	$0$	$- 2$	$- 2$
	\xa0	$- 2$			\xa0
	$1$	$- 1$

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 $(0)$ .

$- 2$	$1$	$1$	$0$	$- 2$	$- 2$
	\xa0	$- 2$	$2$		\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.

$- 2$	$1$	$1$	$0$	$- 2$	$- 2$
	\xa0	$- 2$	$2$		\xa0
	$1$	$- 1$	$2$

Multiply the newest entry in the result $(2)$ by the divisor $(- 2)$ and place the result of $(- 4)$ under the next term in the dividend $(- 2)$ .

$- 2$	$1$	$1$	$0$	$- 2$	$- 2$
	\xa0	$- 2$	$2$	$- 4$	\xa0
	$1$	$- 1$	$2$

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

$- 2$	$1$	$1$	$0$	$- 2$	$- 2$
	\xa0	$- 2$	$2$	$- 4$	\xa0
	$1$	$- 1$	$2$	$- 6$

Multiply the newest entry in the result $(- 6)$ by the divisor $(- 2)$ and place the result of $(12)$ under the next term in the dividend $(- 2)$ .

$- 2$	$1$	$1$	$0$	$- 2$	$- 2$
	\xa0	$- 2$	$2$	$- 4$	$12$
	$1$	$- 1$	$2$	$- 6$

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

$- 2$	$1$	$1$	$0$	$- 2$	$- 2$
	\xa0	$- 2$	$2$	$- 4$	$12$
	$1$	$- 1$	$2$	$- 6$	$10$

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

$1 x^{3} + - 1 x^{2} + (2) x - 6 + \frac{10}{x + 2}$

Simplify the quotient polynomial.

$x^{3} - x^{2} + 2 x - 6 + \frac{10}{x + 2}$

Step 7

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 8

Determine the upper and lower bounds.

Upper Bound: $2$

Lower Bounds: $- 1, - 2$

Step 9

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

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