Examples

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

Step 1

If a polynomial function has integer coefficients, then every rational zero will have the form $\frac{p}{q}$ where $p$ is a factor of the constant and $q$ is a factor of the leading coefficient.

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

$q = \pm 1$

Step 2

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

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

Step 3

Substitute the possible roots one by one into the polynomial to find the actual roots. Simplify to check if the value is $0$ , which means it is a root.

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

Step 4

Simplify the expression. In this case, the expression is equal to $0$ so $x = - 2$ 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 $- 2$ to the power of $2$ .

$4 + 4 (- 2) + 4$

Step 4.1.2

Multiply $4$ by $- 2$ .

$4 - 8 + 4$

Step 4.2

Simplify by adding and subtracting.

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Step 4.2.1

Subtract $8$ from $4$ .

$- 4 + 4$

Step 4.2.2

Add $- 4$ and $4$ .

$0$

Step 5

Since $- 2$ is a known root, divide the polynomial by $x + 2$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

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

Step 6

Next, find the roots of the remaining polynomial. The order of the polynomial has been reduced by $1$ .

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Step 6.1

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

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

Step 6.2

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

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

	$1$

Step 6.3

Multiply the newest entry in the result $(1)$ by the divisor $(- 2)$ and place the result of $(- 2)$ under the next term in the dividend $(4)$ .

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

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.

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

Step 6.5

Multiply the newest entry in the result $(2)$ by the divisor $(- 2)$ and place the result of $(- 4)$ under the next term in the dividend $(4)$ .

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

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.

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

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.

$(1) x + 2$

Step 6.8

Simplify the quotient polynomial.

$x + 2$

Step 7

Subtract $2$ from both sides of the equation.

$x = - 2$

Step 8

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

$x + 2$

Step 9

These are the roots (zeros) of the polynomial $x^{2} + 4 x + 4$ .

$x = - 2$

Step 10

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