Precalculus Examples

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

Step 1

Find every combination of

$\pm \frac{p}{q}$ .

Tap for more steps...

Step 1.1

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.

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

$q = \pm 1$

Step 1.2

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

$\frac{x^{2} - 9}{x - 3}$ when

$x = 3$ .

Tap for more steps...

Step 2.1

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

$3$	$1$	$0$	$- 9$

Step 2.2

The first number in the dividend

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

$3$	$1$	$0$	$- 9$

	$1$

Step 2.3

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

$(0)$ .

$3$	$1$	$0$	$- 9$
		$3$
	$1$

Step 2.4

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

$3$	$1$	$0$	$- 9$
		$3$
	$1$	$3$

Step 2.5

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

$(- 9)$ .

$3$	$1$	$0$	$- 9$
		$3$	$9$
	$1$	$3$

Step 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.

$3$	$1$	$0$	$- 9$
		$3$	$9$
	$1$	$3$	$0$

Step 2.7

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

$(1) x + 3$

Step 2.8

Simplify the quotient polynomial.

$x + 3$

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

$\frac{x^{2} - 9}{x + 3}$ when

$x = - 3$ .

Tap for more steps...

Step 4.1

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

$- 3$	$1$	$0$	$- 9$

Step 4.2

The first number in the dividend

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

$- 3$	$1$	$0$	$- 9$

	$1$

Step 4.3

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

$(0)$ .

$- 3$	$1$	$0$	$- 9$
		$- 3$
	$1$

Step 4.4

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

$- 3$	$1$	$0$	$- 9$
		$- 3$
	$1$	$- 3$

Step 4.5

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

$(- 9)$ .

$- 3$	$1$	$0$	$- 9$
		$- 3$	$9$
	$1$	$- 3$

Step 4.6

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

$- 3$	$1$	$0$	$- 9$
		$- 3$	$9$
	$1$	$- 3$	$0$

Step 4.7

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

$(1) x - 3$

Step 4.8

Simplify the quotient polynomial.

$x - 3$

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

$\frac{x^{2} - 9}{x - 9}$ when

$x = 9$ .

Tap for more steps...

Step 6.1

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

$9$	$1$	$0$	$- 9$

Step 6.2

The first number in the dividend

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

$9$	$1$	$0$	$- 9$

	$1$

Step 6.3

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

$(0)$ .

$9$	$1$	$0$	$- 9$
		$9$
	$1$

Step 6.4

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

$9$	$1$	$0$	$- 9$
		$9$
	$1$	$9$

Step 6.5

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

$(- 9)$ .

$9$	$1$	$0$	$- 9$
		$9$	$81$
	$1$	$9$

Step 6.6

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

$9$	$1$	$0$	$- 9$
		$9$	$81$
	$1$	$9$	$72$

Step 6.7

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

$(1) x + 9 + \frac{72}{x - 9}$

Step 6.8

Simplify the quotient polynomial.

$x + 9 + \frac{72}{x - 9}$

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

$\frac{x^{2} - 9}{x + 9}$ when

$x = - 9$ .

Tap for more steps...

Step 8.1

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

$- 9$	$1$	$0$	$- 9$

Step 8.2

The first number in the dividend

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

$- 9$	$1$	$0$	$- 9$

	$1$

Step 8.3

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

$(0)$ .

$- 9$	$1$	$0$	$- 9$
		$- 9$
	$1$

Step 8.4

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

$- 9$	$1$	$0$	$- 9$
		$- 9$
	$1$	$- 9$

Step 8.5

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

$(- 9)$ .

$- 9$	$1$	$0$	$- 9$
		$- 9$	$81$
	$1$	$- 9$

Step 8.6

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

$- 9$	$1$	$0$	$- 9$
		$- 9$	$81$
	$1$	$- 9$	$72$

Step 8.7

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

$(1) x - 9 + \frac{72}{x + 9}$

Step 8.8

Simplify the quotient polynomial.

$x - 9 + \frac{72}{x + 9}$

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

Enter YOUR Problem