Find the Upper and Lower Bounds f(x)=x^4-8x^2-9

$f (x) = x^{4} - 8 x^{2} - 9$

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.

$p = \pm 1, \pm 3, \pm 9$

$q = \pm 1$

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

$\pm 1, \pm 3, \pm 9$

Step 2

Apply synthetic division on $\frac{x^{4} - 8 x^{2} - 9}{x - 3}$ when $x = 3$ .

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

$3$	$1$	$0$	$- 8$	$0$	$- 9$
	\xa0				\xa0

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

$3$	$1$	$0$	$- 8$	$0$	$- 9$
	\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 $(0)$ .

$3$	$1$	$0$	$- 8$	$0$	$- 9$
	\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.

$3$	$1$	$0$	$- 8$	$0$	$- 9$
	\xa0	$3$			\xa0
	$1$	$3$

Multiply the newest entry in the result $(3)$ by the divisor $(3)$ and place the result of $(9)$ under the next term in the dividend $(- 8)$ .

$3$	$1$	$0$	$- 8$	$0$	$- 9$
	\xa0	$3$	$9$		\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.

$3$	$1$	$0$	$- 8$	$0$	$- 9$
	\xa0	$3$	$9$		\xa0
	$1$	$3$	$1$

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 $(0)$ .

$3$	$1$	$0$	$- 8$	$0$	$- 9$
	\xa0	$3$	$9$	$3$	\xa0
	$1$	$3$	$1$

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

$3$	$1$	$0$	$- 8$	$0$	$- 9$
	\xa0	$3$	$9$	$3$	\xa0
	$1$	$3$	$1$	$3$

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

$3$	$1$	$0$	$- 8$	$0$	$- 9$
	\xa0	$3$	$9$	$3$	$9$
	$1$	$3$	$1$	$3$

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

$3$	$1$	$0$	$- 8$	$0$	$- 9$
	\xa0	$3$	$9$	$3$	$9$
	$1$	$3$	$1$	$3$	$0$

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

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

Simplify the quotient polynomial.

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

Step 3

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 4

Apply synthetic division on $\frac{x^{4} - 8 x^{2} - 9}{x + 3}$ when $x = - 3$ .

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

$- 3$	$1$	$0$	$- 8$	$0$	$- 9$
	\xa0				\xa0

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

$- 3$	$1$	$0$	$- 8$	$0$	$- 9$
	\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 $(0)$ .

$- 3$	$1$	$0$	$- 8$	$0$	$- 9$
	\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.

$- 3$	$1$	$0$	$- 8$	$0$	$- 9$
	\xa0	$- 3$			\xa0
	$1$	$- 3$

Multiply the newest entry in the result $(- 3)$ by the divisor $(- 3)$ and place the result of $(9)$ under the next term in the dividend $(- 8)$ .

$- 3$	$1$	$0$	$- 8$	$0$	$- 9$
	\xa0	$- 3$	$9$		\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.

$- 3$	$1$	$0$	$- 8$	$0$	$- 9$
	\xa0	$- 3$	$9$		\xa0
	$1$	$- 3$	$1$

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 $(0)$ .

$- 3$	$1$	$0$	$- 8$	$0$	$- 9$
	\xa0	$- 3$	$9$	$- 3$	\xa0
	$1$	$- 3$	$1$

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

$- 3$	$1$	$0$	$- 8$	$0$	$- 9$
	\xa0	$- 3$	$9$	$- 3$	\xa0
	$1$	$- 3$	$1$	$- 3$

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

$- 3$	$1$	$0$	$- 8$	$0$	$- 9$
	\xa0	$- 3$	$9$	$- 3$	$9$
	$1$	$- 3$	$1$	$- 3$

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

$- 3$	$1$	$0$	$- 8$	$0$	$- 9$
	\xa0	$- 3$	$9$	$- 3$	$9$
	$1$	$- 3$	$1$	$- 3$	$0$

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

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

Simplify the quotient polynomial.

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

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 $\frac{x^{4} - 8 x^{2} - 9}{x - 9}$ when $x = 9$ .

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

$9$	$1$	$0$	$- 8$	$0$	$- 9$
	\xa0				\xa0

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

$9$	$1$	$0$	$- 8$	$0$	$- 9$
	\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 $(0)$ .

$9$	$1$	$0$	$- 8$	$0$	$- 9$
	\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.

$9$	$1$	$0$	$- 8$	$0$	$- 9$
	\xa0	$9$			\xa0
	$1$	$9$

Multiply the newest entry in the result $(9)$ by the divisor $(9)$ and place the result of $(81)$ under the next term in the dividend $(- 8)$ .

$9$	$1$	$0$	$- 8$	$0$	$- 9$
	\xa0	$9$	$81$		\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.

$9$	$1$	$0$	$- 8$	$0$	$- 9$
	\xa0	$9$	$81$		\xa0
	$1$	$9$	$73$

Multiply the newest entry in the result $(73)$ by the divisor $(9)$ and place the result of $(657)$ under the next term in the dividend $(0)$ .

$9$	$1$	$0$	$- 8$	$0$	$- 9$
	\xa0	$9$	$81$	$657$	\xa0
	$1$	$9$	$73$

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

$9$	$1$	$0$	$- 8$	$0$	$- 9$
	\xa0	$9$	$81$	$657$	\xa0
	$1$	$9$	$73$	$657$

Multiply the newest entry in the result $(657)$ by the divisor $(9)$ and place the result of $(5913)$ under the next term in the dividend $(- 9)$ .

$9$	$1$	$0$	$- 8$	$0$	$- 9$
	\xa0	$9$	$81$	$657$	$5913$
	$1$	$9$	$73$	$657$

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

$9$	$1$	$0$	$- 8$	$0$	$- 9$
	\xa0	$9$	$81$	$657$	$5913$
	$1$	$9$	$73$	$657$	$5904$

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

$1 x^{3} + 9 x^{2} + (73) x + 657 + \frac{5904}{x - 9}$

Simplify the quotient polynomial.

$x^{3} + 9 x^{2} + 73 x + 657 + \frac{5904}{x - 9}$

Step 7

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 8

Apply synthetic division on $\frac{x^{4} - 8 x^{2} - 9}{x + 9}$ when $x = - 9$ .

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

$- 9$	$1$	$0$	$- 8$	$0$	$- 9$
	\xa0				\xa0

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

$- 9$	$1$	$0$	$- 8$	$0$	$- 9$
	\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 $(0)$ .

$- 9$	$1$	$0$	$- 8$	$0$	$- 9$
	\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.

$- 9$	$1$	$0$	$- 8$	$0$	$- 9$
	\xa0	$- 9$			\xa0
	$1$	$- 9$

Multiply the newest entry in the result $(- 9)$ by the divisor $(- 9)$ and place the result of $(81)$ under the next term in the dividend $(- 8)$ .

$- 9$	$1$	$0$	$- 8$	$0$	$- 9$
	\xa0	$- 9$	$81$		\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.

$- 9$	$1$	$0$	$- 8$	$0$	$- 9$
	\xa0	$- 9$	$81$		\xa0
	$1$	$- 9$	$73$

Multiply the newest entry in the result $(73)$ by the divisor $(- 9)$ and place the result of $(- 657)$ under the next term in the dividend $(0)$ .

$- 9$	$1$	$0$	$- 8$	$0$	$- 9$
	\xa0	$- 9$	$81$	$- 657$	\xa0
	$1$	$- 9$	$73$

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

$- 9$	$1$	$0$	$- 8$	$0$	$- 9$
	\xa0	$- 9$	$81$	$- 657$	\xa0
	$1$	$- 9$	$73$	$- 657$

Multiply the newest entry in the result $(- 657)$ by the divisor $(- 9)$ and place the result of $(5913)$ under the next term in the dividend $(- 9)$ .

$- 9$	$1$	$0$	$- 8$	$0$	$- 9$
	\xa0	$- 9$	$81$	$- 657$	$5913$
	$1$	$- 9$	$73$	$- 657$

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

$- 9$	$1$	$0$	$- 8$	$0$	$- 9$
	\xa0	$- 9$	$81$	$- 657$	$5913$
	$1$	$- 9$	$73$	$- 657$	$5904$

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

$1 x^{3} + - 9 x^{2} + (73) x - 657 + \frac{5904}{x + 9}$

Simplify the quotient polynomial.

$x^{3} - 9 x^{2} + 73 x - 657 + \frac{5904}{x + 9}$

Step 9

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 10

Determine the upper and lower bounds.

Upper Bounds: $3, 9$

Lower Bounds: $- 3, - 9$

Step 11

Find the Upper and Lower Bounds f(x)=x^4-8x^2-9

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