Examples

$x^{2} - 5 x + 6$

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.

$p = \pm 1, \pm 2, \pm 3, \pm 6$

$q = \pm 1$

Step 2

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

$\pm 1, \pm 2, \pm 3, \pm 6$

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.

${(2)}^{2} - 5 \cdot 2 + 6$

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 $2$ .

$4 - 5 \cdot 2 + 6$

Step 4.1.2

Multiply $- 5$ by $2$ .

$4 - 10 + 6$

Step 4.2

Simplify by adding and subtracting.

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Step 4.2.1

Subtract $10$ from $4$ .

$- 6 + 6$

Step 4.2.2

Add $- 6$ and $6$ .

$0$

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.

$\frac{x^{2} - 5 x + 6}{x - 2}$

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.

$2$	$1$	$- 5$	$6$

Step 6.2

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

$2$	$1$	$- 5$	$6$

	$1$

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 $(- 5)$ .

$2$	$1$	$- 5$	$6$
		$2$
	$1$

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.

$2$	$1$	$- 5$	$6$
		$2$
	$1$	$- 3$

Step 6.5

Multiply the newest entry in the result $(- 3)$ by the divisor $(2)$ and place the result of $(- 6)$ under the next term in the dividend $(6)$ .

$2$	$1$	$- 5$	$6$
		$2$	$- 6$
	$1$	$- 3$

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.

$2$	$1$	$- 5$	$6$
		$2$	$- 6$
	$1$	$- 3$	$0$

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.

$(1) x - 3$

Step 6.8

Simplify the quotient polynomial.

$x - 3$

Step 7

Add $3$ to both sides of the equation.

$x = 3$

Step 8

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

$(x - 2) (x - 3)$

Step 9

These are the roots (zeros) of the polynomial $x^{2} - 5 x + 6$ .

$x = 2, 3$

Step 10

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