Precalculus Examples

f (x) = x^{2} - 5

$f (x) = x^{2} - 5$

Step 1

Find every combination of

\pm \frac{p}{q}

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

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Step 1.1

If a polynomial function has integer coefficients, then every rational zero will have the form

\frac{p}{q}

$\frac{p}{q}$ where

p

$p$ is a factor of the constant and

q

$q$ is a factor of the leading coefficient.

p = \pm 1, \pm 5

$p = \pm 1, \pm 5$

q = \pm 1

$q = \pm 1$

Step 1.2

Find every combination of

\pm \frac{p}{q}

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

\pm 1, \pm 5

$\pm 1, \pm 5$

\pm 1, \pm 5

$\pm 1, \pm 5$

Step 2

Apply synthetic division on

\frac{x^{2} - 5}{x - 5}

$\frac{x^{2} - 5}{x - 5}$ when

x = 5

$x = 5$ .

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Step 2.1

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

$5$ $5$	$1$ $1$	$0$ $0$	$- 5$ $- 5$

Step 2.2

The first number in the dividend

(1)

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

$5$ $5$	$1$ $1$	$0$ $0$	$- 5$ $- 5$

	$1$ $1$

Step 2.3

Multiply the newest entry in the result

(1)

$(1)$ by the divisor

(5)

$(5)$ and place the result of

(5)

$(5)$ under the next term in the dividend

(0)

$(0)$ .

$5$ $5$	$1$ $1$	$0$ $0$	$- 5$ $- 5$
		$5$ $5$
	$1$ $1$

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.

$5$ $5$	$1$ $1$	$0$ $0$	$- 5$ $- 5$
		$5$ $5$
	$1$ $1$	$5$ $5$

Step 2.5

Multiply the newest entry in the result

(5)

$(5)$ by the divisor

(5)

$(5)$ and place the result of

(25)

$(25)$ under the next term in the dividend

(- 5)

$(- 5)$ .

$5$ $5$	$1$ $1$	$0$ $0$	$- 5$ $- 5$
		$5$ $5$	$25$ $25$
	$1$ $1$	$5$ $5$

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.

$5$ $5$	$1$ $1$	$0$ $0$	$- 5$ $- 5$
		$5$ $5$	$25$ $25$
	$1$ $1$	$5$ $5$	$20$ $20$

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 + 5 + \frac{20}{x - 5}

$(1) x + 5 + \frac{20}{x - 5}$

Step 2.8

Simplify the quotient polynomial.

x + 5 + \frac{20}{x - 5}

$x + 5 + \frac{20}{x - 5}$

x + 5 + \frac{20}{x - 5}

$x + 5 + \frac{20}{x - 5}$

Step 3

Since

5 > 0

$5 > 0$ and all of the signs in the bottom row of the synthetic division are positive,

5

$5$ is an upper bound for the real roots of the function.

Upper Bound:

5

$5$

Step 4

Apply synthetic division on

\frac{x^{2} - 5}{x + 5}

$\frac{x^{2} - 5}{x + 5}$ when

x = - 5

$x = - 5$ .

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Step 4.1

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

$- 5$ $- 5$	$1$ $1$	$0$ $0$	$- 5$ $- 5$

Step 4.2

The first number in the dividend

(1)

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

$- 5$ $- 5$	$1$ $1$	$0$ $0$	$- 5$ $- 5$

	$1$ $1$

Step 4.3

Multiply the newest entry in the result

(1)

$(1)$ by the divisor

(- 5)

$(- 5)$ and place the result of

(- 5)

$(- 5)$ under the next term in the dividend

(0)

$(0)$ .

$- 5$ $- 5$	$1$ $1$	$0$ $0$	$- 5$ $- 5$
		$- 5$ $- 5$
	$1$ $1$

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.

$- 5$ $- 5$	$1$ $1$	$0$ $0$	$- 5$ $- 5$
		$- 5$ $- 5$
	$1$ $1$	$- 5$ $- 5$

Step 4.5

Multiply the newest entry in the result

(- 5)

$(- 5)$ by the divisor

(- 5)

$(- 5)$ and place the result of

(25)

$(25)$ under the next term in the dividend

(- 5)

$(- 5)$ .

$- 5$ $- 5$	$1$ $1$	$0$ $0$	$- 5$ $- 5$
		$- 5$ $- 5$	$25$ $25$
	$1$ $1$	$- 5$ $- 5$

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.

$- 5$ $- 5$	$1$ $1$	$0$ $0$	$- 5$ $- 5$
		$- 5$ $- 5$	$25$ $25$
	$1$ $1$	$- 5$ $- 5$	$20$ $20$

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 - 5 + \frac{20}{x + 5}

$(1) x - 5 + \frac{20}{x + 5}$

Step 4.8

Simplify the quotient polynomial.

x - 5 + \frac{20}{x + 5}

$x - 5 + \frac{20}{x + 5}$

x - 5 + \frac{20}{x + 5}

$x - 5 + \frac{20}{x + 5}$

Step 5

Since

- 5 < 0

$- 5 < 0$ and the signs in the bottom row of the synthetic division alternate sign,

- 5

$- 5$ is a lower bound for the real roots of the function.

Lower Bound:

- 5

$- 5$

Step 6

Determine the upper and lower bounds.

Upper Bound:

5

$5$

Lower Bound:

- 5

$- 5$

Step 7

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