Examples

x^{2} + 4 x + 4

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

Step 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 2, \pm 4

$p = \pm 1, \pm 2, \pm 4$

q = \pm 1

$q = \pm 1$

Step 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 2, \pm 4

$\pm 1, \pm 2, \pm 4$

Step 3

Substitute the possible roots one by one into the polynomial to find the actual roots. Simplify to check if the value is

0

$0$ , which means it is a root.

{(- 2)}^{2} + 4 (- 2) + 4

${(- 2)}^{2} + 4 (- 2) + 4$

Step 4

Simplify the expression. In this case, the expression is equal to

0

$0$ so

x = - 2

$x = - 2$ is a root of the polynomial.

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Step 4.1

Simplify each term.

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Step 4.1.1

Raise

- 2

$- 2$ to the power of

2

$2$ .

4 + 4 (- 2) + 4

$4 + 4 (- 2) + 4$

Step 4.1.2

Multiply

4

$4$ by

- 2

$- 2$ .

4 - 8 + 4

$4 - 8 + 4$

4 - 8 + 4

$4 - 8 + 4$

Step 4.2

Simplify by adding and subtracting.

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Step 4.2.1

Subtract

8

$8$ from

4

$4$ .

- 4 + 4

$- 4 + 4$

Step 4.2.2

Add

- 4

$- 4$ and

4

$4$ .

0

$0$

0

$0$

0

$0$

Step 5

Since

- 2

$- 2$ is a known root, divide the polynomial by

x + 2

$x + 2$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

\frac{x^{2} + 4 x + 4}{x + 2}

$\frac{x^{2} + 4 x + 4}{x + 2}$

Step 6

Next, find the roots of the remaining polynomial. The order of the polynomial has been reduced by

1

$1$ .

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Step 6.1

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

$- 2$ $- 2$	$1$ $1$	$4$ $4$	$4$ $4$

Step 6.2

The first number in the dividend

(1)

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

$- 2$ $- 2$	$1$ $1$	$4$ $4$	$4$ $4$

	$1$ $1$

Step 6.3

Multiply the newest entry in the result

(1)

$(1)$ by the divisor

(- 2)

$(- 2)$ and place the result of

(- 2)

$(- 2)$ under the next term in the dividend

(4)

$(4)$ .

$- 2$ $- 2$	$1$ $1$	$4$ $4$	$4$ $4$
		$- 2$ $- 2$
	$1$ $1$

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.

$- 2$ $- 2$	$1$ $1$	$4$ $4$	$4$ $4$
		$- 2$ $- 2$
	$1$ $1$	$2$ $2$

Step 6.5

Multiply the newest entry in the result

(2)

$(2)$ by the divisor

(- 2)

$(- 2)$ and place the result of

(- 4)

$(- 4)$ under the next term in the dividend

(4)

$(4)$ .

$- 2$ $- 2$	$1$ $1$	$4$ $4$	$4$ $4$
		$- 2$ $- 2$	$- 4$ $- 4$
	$1$ $1$	$2$ $2$

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.

$- 2$ $- 2$	$1$ $1$	$4$ $4$	$4$ $4$
		$- 2$ $- 2$	$- 4$ $- 4$
	$1$ $1$	$2$ $2$	$0$ $0$

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 + 2

$(1) x + 2$

Step 6.8

Simplify the quotient polynomial.

x + 2

$x + 2$

x + 2

$x + 2$

Step 7

Subtract

2

$2$ from both sides of the equation.

x = - 2

$x = - 2$

Step 8

The polynomial can be written as a set of linear factors.

x + 2

$x + 2$

Step 9

These are the roots (zeros) of the polynomial

x^{2} + 4 x + 4

$x^{2} + 4 x + 4$ .

x = - 2

$x = - 2$

Step 10

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