Find the Upper and Lower Bounds f(x)=x^4-24x^2-25

$f (x) = x^{4} - 24 x^{2} - 25$

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.

$p = \pm 1, \pm 5, \pm 25$

$q = \pm 1$

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

$\pm 1, \pm 5, \pm 25$

Step 2

Apply synthetic division on $\frac{x^{4} - 24 x^{2} - 25}{x - 5}$ when $x = 5$ .

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

$5$	$1$	$0$	$- 24$	$0$	$- 25$
	\xa0				\xa0

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

$5$	$1$	$0$	$- 24$	$0$	$- 25$
	\xa0				\xa0
	$1$

Multiply the newest entry in the result $(1)$ by the divisor $(5)$ and place the result of $(5)$ under the next term in the dividend $(0)$ .

$5$	$1$	$0$	$- 24$	$0$	$- 25$
	\xa0	$5$			\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.

$5$	$1$	$0$	$- 24$	$0$	$- 25$
	\xa0	$5$			\xa0
	$1$	$5$

Multiply the newest entry in the result $(5)$ by the divisor $(5)$ and place the result of $(25)$ under the next term in the dividend $(- 24)$ .

$5$	$1$	$0$	$- 24$	$0$	$- 25$
	\xa0	$5$	$25$		\xa0
	$1$	$5$

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

$5$	$1$	$0$	$- 24$	$0$	$- 25$
	\xa0	$5$	$25$		\xa0
	$1$	$5$	$1$

Multiply the newest entry in the result $(1)$ by the divisor $(5)$ and place the result of $(5)$ under the next term in the dividend $(0)$ .

$5$	$1$	$0$	$- 24$	$0$	$- 25$
	\xa0	$5$	$25$	$5$	\xa0
	$1$	$5$	$1$

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

$5$	$1$	$0$	$- 24$	$0$	$- 25$
	\xa0	$5$	$25$	$5$	\xa0
	$1$	$5$	$1$	$5$

Multiply the newest entry in the result $(5)$ by the divisor $(5)$ and place the result of $(25)$ under the next term in the dividend $(- 25)$ .

$5$	$1$	$0$	$- 24$	$0$	$- 25$
	\xa0	$5$	$25$	$5$	$25$
	$1$	$5$	$1$	$5$

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

$5$	$1$	$0$	$- 24$	$0$	$- 25$
	\xa0	$5$	$25$	$5$	$25$
	$1$	$5$	$1$	$5$	$0$

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

$1 x^{3} + 5 x^{2} + (1) x + 5$

Simplify the quotient polynomial.

$x^{3} + 5 x^{2} + x + 5$

Step 3

Since $5 > 0$ and all of the signs in the bottom row of the synthetic division are positive, $5$ is an upper bound for the real roots of the function.

Upper Bound: $5$

Step 4

Apply synthetic division on $\frac{x^{4} - 24 x^{2} - 25}{x + 5}$ when $x = - 5$ .

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

$- 5$	$1$	$0$	$- 24$	$0$	$- 25$
	\xa0				\xa0

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

$- 5$	$1$	$0$	$- 24$	$0$	$- 25$
	\xa0				\xa0
	$1$

Multiply the newest entry in the result $(1)$ by the divisor $(- 5)$ and place the result of $(- 5)$ under the next term in the dividend $(0)$ .

$- 5$	$1$	$0$	$- 24$	$0$	$- 25$
	\xa0	$- 5$			\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.

$- 5$	$1$	$0$	$- 24$	$0$	$- 25$
	\xa0	$- 5$			\xa0
	$1$	$- 5$

Multiply the newest entry in the result $(- 5)$ by the divisor $(- 5)$ and place the result of $(25)$ under the next term in the dividend $(- 24)$ .

$- 5$	$1$	$0$	$- 24$	$0$	$- 25$
	\xa0	$- 5$	$25$		\xa0
	$1$	$- 5$

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

$- 5$	$1$	$0$	$- 24$	$0$	$- 25$
	\xa0	$- 5$	$25$		\xa0
	$1$	$- 5$	$1$

Multiply the newest entry in the result $(1)$ by the divisor $(- 5)$ and place the result of $(- 5)$ under the next term in the dividend $(0)$ .

$- 5$	$1$	$0$	$- 24$	$0$	$- 25$
	\xa0	$- 5$	$25$	$- 5$	\xa0
	$1$	$- 5$	$1$

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

$- 5$	$1$	$0$	$- 24$	$0$	$- 25$
	\xa0	$- 5$	$25$	$- 5$	\xa0
	$1$	$- 5$	$1$	$- 5$

Multiply the newest entry in the result $(- 5)$ by the divisor $(- 5)$ and place the result of $(25)$ under the next term in the dividend $(- 25)$ .

$- 5$	$1$	$0$	$- 24$	$0$	$- 25$
	\xa0	$- 5$	$25$	$- 5$	$25$
	$1$	$- 5$	$1$	$- 5$

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

$- 5$	$1$	$0$	$- 24$	$0$	$- 25$
	\xa0	$- 5$	$25$	$- 5$	$25$
	$1$	$- 5$	$1$	$- 5$	$0$

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

$1 x^{3} + - 5 x^{2} + (1) x - 5$

Simplify the quotient polynomial.

$x^{3} - 5 x^{2} + x - 5$

Step 5

Since $- 5 < 0$ and the signs in the bottom row of the synthetic division alternate sign, $- 5$ is a lower bound for the real roots of the function.

Lower Bound: $- 5$

Step 6

Apply synthetic division on $\frac{x^{4} - 24 x^{2} - 25}{x - 25}$ when $x = 25$ .

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

$25$	$1$	$0$	$- 24$	$0$	$- 25$
	\xa0				\xa0

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

$25$	$1$	$0$	$- 24$	$0$	$- 25$
	\xa0				\xa0
	$1$

Multiply the newest entry in the result $(1)$ by the divisor $(25)$ and place the result of $(25)$ under the next term in the dividend $(0)$ .

$25$	$1$	$0$	$- 24$	$0$	$- 25$
	\xa0	$25$			\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.

$25$	$1$	$0$	$- 24$	$0$	$- 25$
	\xa0	$25$			\xa0
	$1$	$25$

Multiply the newest entry in the result $(25)$ by the divisor $(25)$ and place the result of $(625)$ under the next term in the dividend $(- 24)$ .

$25$	$1$	$0$	$- 24$	$0$	$- 25$
	\xa0	$25$	$625$		\xa0
	$1$	$25$

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

$25$	$1$	$0$	$- 24$	$0$	$- 25$
	\xa0	$25$	$625$		\xa0
	$1$	$25$	$601$

Multiply the newest entry in the result $(601)$ by the divisor $(25)$ and place the result of $(15025)$ under the next term in the dividend $(0)$ .

$25$	$1$	$0$	$- 24$	$0$	$- 25$
	\xa0	$25$	$625$	$15025$	\xa0
	$1$	$25$	$601$

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

$25$	$1$	$0$	$- 24$	$0$	$- 25$
	\xa0	$25$	$625$	$15025$	\xa0
	$1$	$25$	$601$	$15025$

Multiply the newest entry in the result $(15025)$ by the divisor $(25)$ and place the result of $(375625)$ under the next term in the dividend $(- 25)$ .

$25$	$1$	$0$	$- 24$	$0$	$- 25$
	\xa0	$25$	$625$	$15025$	$375625$
	$1$	$25$	$601$	$15025$

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

$25$	$1$	$0$	$- 24$	$0$	$- 25$
	\xa0	$25$	$625$	$15025$	$375625$
	$1$	$25$	$601$	$15025$	$375600$

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

$1 x^{3} + 25 x^{2} + (601) x + 15025 + \frac{375600}{x - 25}$

Simplify the quotient polynomial.

$x^{3} + 25 x^{2} + 601 x + 15025 + \frac{375600}{x - 25}$

Step 7

Since $25 > 0$ and all of the signs in the bottom row of the synthetic division are positive, $25$ is an upper bound for the real roots of the function.

Upper Bound: $25$

Step 8

Apply synthetic division on $\frac{x^{4} - 24 x^{2} - 25}{x + 25}$ when $x = - 25$ .

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

$- 25$	$1$	$0$	$- 24$	$0$	$- 25$
	\xa0				\xa0

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

$- 25$	$1$	$0$	$- 24$	$0$	$- 25$
	\xa0				\xa0
	$1$

Multiply the newest entry in the result $(1)$ by the divisor $(- 25)$ and place the result of $(- 25)$ under the next term in the dividend $(0)$ .

$- 25$	$1$	$0$	$- 24$	$0$	$- 25$
	\xa0	$- 25$			\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.

$- 25$	$1$	$0$	$- 24$	$0$	$- 25$
	\xa0	$- 25$			\xa0
	$1$	$- 25$

Multiply the newest entry in the result $(- 25)$ by the divisor $(- 25)$ and place the result of $(625)$ under the next term in the dividend $(- 24)$ .

$- 25$	$1$	$0$	$- 24$	$0$	$- 25$
	\xa0	$- 25$	$625$		\xa0
	$1$	$- 25$

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

$- 25$	$1$	$0$	$- 24$	$0$	$- 25$
	\xa0	$- 25$	$625$		\xa0
	$1$	$- 25$	$601$

Multiply the newest entry in the result $(601)$ by the divisor $(- 25)$ and place the result of $(- 15025)$ under the next term in the dividend $(0)$ .

$- 25$	$1$	$0$	$- 24$	$0$	$- 25$
	\xa0	$- 25$	$625$	$- 15025$	\xa0
	$1$	$- 25$	$601$

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

$- 25$	$1$	$0$	$- 24$	$0$	$- 25$
	\xa0	$- 25$	$625$	$- 15025$	\xa0
	$1$	$- 25$	$601$	$- 15025$

Multiply the newest entry in the result $(- 15025)$ by the divisor $(- 25)$ and place the result of $(375625)$ under the next term in the dividend $(- 25)$ .

$- 25$	$1$	$0$	$- 24$	$0$	$- 25$
	\xa0	$- 25$	$625$	$- 15025$	$375625$
	$1$	$- 25$	$601$	$- 15025$

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

$- 25$	$1$	$0$	$- 24$	$0$	$- 25$
	\xa0	$- 25$	$625$	$- 15025$	$375625$
	$1$	$- 25$	$601$	$- 15025$	$375600$

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

$1 x^{3} + - 25 x^{2} + (601) x - 15025 + \frac{375600}{x + 25}$

Simplify the quotient polynomial.

$x^{3} - 25 x^{2} + 601 x - 15025 + \frac{375600}{x + 25}$

Step 9

Since $- 25 < 0$ and the signs in the bottom row of the synthetic division alternate sign, $- 25$ is a lower bound for the real roots of the function.

Lower Bound: $- 25$

Step 10

Determine the upper and lower bounds.

Upper Bounds: $5, 25$

Lower Bounds: $- 5, - 25$

Step 11

Find the Upper and Lower Bounds f(x)=x^4-24x^2-25

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