Precalculus Examples

$x^{3} + 10 x^{2} + 23 x + 14$

Step 1

Factor $x^{3} + 10 x^{2} + 23 x + 14$ using the rational roots test.

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Step 1.1

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

$p = \pm 1, \pm 14, \pm 2, \pm 7$

$q = \pm 1$

Step 1.2

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

$\pm 1, \pm 14, \pm 2, \pm 7$

Step 1.3

Substitute $- 1$ and simplify the expression. In this case, the expression is equal to $0$ so $- 1$ is a root of the polynomial.

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Step 1.3.1

Substitute $- 1$ into the polynomial.

${(- 1)}^{3} + 10 {(- 1)}^{2} + 23 \cdot - 1 + 14$

Step 1.3.2

Raise $- 1$ to the power of $3$ .

$- 1 + 10 {(- 1)}^{2} + 23 \cdot - 1 + 14$

Step 1.3.3

Raise $- 1$ to the power of $2$ .

$- 1 + 10 \cdot 1 + 23 \cdot - 1 + 14$

Step 1.3.4

Multiply $10$ by $1$ .

$- 1 + 10 + 23 \cdot - 1 + 14$

Step 1.3.5

Add $- 1$ and $10$ .

$9 + 23 \cdot - 1 + 14$

Step 1.3.6

Multiply $23$ by $- 1$ .

$9 - 23 + 14$

Step 1.3.7

Subtract $23$ from $9$ .

$- 14 + 14$

Step 1.3.8

Add $- 14$ and $14$ .

$0$

Step 1.4

Since $- 1$ is a known root, divide the polynomial by $x + 1$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{x^{3} + 10 x^{2} + 23 x + 14}{x + 1}$

Step 1.5

Divide $x^{3} + 10 x^{2} + 23 x + 14$ by $x + 1$ .

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Step 1.5.1

Set up the polynomials to be divided. If there is not a term for every exponent, insert one with a value of $0$ .

$x$

$1$

$x^{3}$

$10 x^{2}$

$23 x$

$14$

Step 1.5.2

Divide the highest order term in the dividend $x^{3}$ by the highest order term in divisor $x$ .

					$x^{2}$
	$x$	+	$1$		$x^{3}$	+	$10 x^{2}$	+	$23 x$	+	$14$

Step 1.5.3

Multiply the new quotient term by the divisor.

				$x^{2}$
$x$	+	$1$		$x^{3}$	+	$10 x^{2}$	+	$23 x$	+	$14$
			+	$x^{3}$	+	$x^{2}$

Step 1.5.4

The expression needs to be subtracted from the dividend, so change all the signs in $x^{3} + x^{2}$

				$x^{2}$
$x$	+	$1$		$x^{3}$	+	$10 x^{2}$	+	$23 x$	+	$14$
			-	$x^{3}$	-	$x^{2}$

Step 1.5.5

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

				$x^{2}$
$x$	+	$1$		$x^{3}$	+	$10 x^{2}$	+	$23 x$	+	$14$
			-	$x^{3}$	-	$x^{2}$
					+	$9 x^{2}$

Step 1.5.6

Pull the next terms from the original dividend down into the current dividend.

				$x^{2}$
$x$	+	$1$		$x^{3}$	+	$10 x^{2}$	+	$23 x$	+	$14$
			-	$x^{3}$	-	$x^{2}$
					+	$9 x^{2}$	+	$23 x$

Step 1.5.7

Divide the highest order term in the dividend $9 x^{2}$ by the highest order term in divisor $x$ .

				$x^{2}$	+	$9 x$
$x$	+	$1$		$x^{3}$	+	$10 x^{2}$	+	$23 x$	+	$14$
			-	$x^{3}$	-	$x^{2}$
					+	$9 x^{2}$	+	$23 x$

Step 1.5.8

Multiply the new quotient term by the divisor.

				$x^{2}$	+	$9 x$
$x$	+	$1$		$x^{3}$	+	$10 x^{2}$	+	$23 x$	+	$14$
			-	$x^{3}$	-	$x^{2}$
					+	$9 x^{2}$	+	$23 x$
					+	$9 x^{2}$	+	$9 x$

Step 1.5.9

The expression needs to be subtracted from the dividend, so change all the signs in $9 x^{2} + 9 x$

				$x^{2}$	+	$9 x$
$x$	+	$1$		$x^{3}$	+	$10 x^{2}$	+	$23 x$	+	$14$
			-	$x^{3}$	-	$x^{2}$
					+	$9 x^{2}$	+	$23 x$
					-	$9 x^{2}$	-	$9 x$

Step 1.5.10

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

				$x^{2}$	+	$9 x$
$x$	+	$1$		$x^{3}$	+	$10 x^{2}$	+	$23 x$	+	$14$
			-	$x^{3}$	-	$x^{2}$
					+	$9 x^{2}$	+	$23 x$
					-	$9 x^{2}$	-	$9 x$
							+	$14 x$

Step 1.5.11

Pull the next terms from the original dividend down into the current dividend.

				$x^{2}$	+	$9 x$
$x$	+	$1$		$x^{3}$	+	$10 x^{2}$	+	$23 x$	+	$14$
			-	$x^{3}$	-	$x^{2}$
					+	$9 x^{2}$	+	$23 x$
					-	$9 x^{2}$	-	$9 x$
							+	$14 x$	+	$14$

Step 1.5.12

Divide the highest order term in the dividend $14 x$ by the highest order term in divisor $x$ .

				$x^{2}$	+	$9 x$	+	$14$
$x$	+	$1$		$x^{3}$	+	$10 x^{2}$	+	$23 x$	+	$14$
			-	$x^{3}$	-	$x^{2}$
					+	$9 x^{2}$	+	$23 x$
					-	$9 x^{2}$	-	$9 x$
							+	$14 x$	+	$14$

Step 1.5.13

Multiply the new quotient term by the divisor.

				$x^{2}$	+	$9 x$	+	$14$
$x$	+	$1$		$x^{3}$	+	$10 x^{2}$	+	$23 x$	+	$14$
			-	$x^{3}$	-	$x^{2}$
					+	$9 x^{2}$	+	$23 x$
					-	$9 x^{2}$	-	$9 x$
							+	$14 x$	+	$14$
							+	$14 x$	+	$14$

Step 1.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $14 x + 14$

				$x^{2}$	+	$9 x$	+	$14$
$x$	+	$1$		$x^{3}$	+	$10 x^{2}$	+	$23 x$	+	$14$
			-	$x^{3}$	-	$x^{2}$
					+	$9 x^{2}$	+	$23 x$
					-	$9 x^{2}$	-	$9 x$
							+	$14 x$	+	$14$
							-	$14 x$	-	$14$

Step 1.5.15

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

				$x^{2}$	+	$9 x$	+	$14$
$x$	+	$1$		$x^{3}$	+	$10 x^{2}$	+	$23 x$	+	$14$
			-	$x^{3}$	-	$x^{2}$
					+	$9 x^{2}$	+	$23 x$
					-	$9 x^{2}$	-	$9 x$
							+	$14 x$	+	$14$
							-	$14 x$	-	$14$
										$0$

Step 1.5.16

Since the remander is $0$ , the final answer is the quotient.

$x^{2} + 9 x + 14$

Step 1.6

Write $x^{3} + 10 x^{2} + 23 x + 14$ as a set of factors.

$(x + 1) (x^{2} + 9 x + 14)$

Step 2

Factor $x^{2} + 9 x + 14$ using the AC method.

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Step 2.1

Factor $x^{2} + 9 x + 14$ using the AC method.

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Step 2.1.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $14$ and whose sum is $9$ .

$2, 7$

Step 2.1.2

Write the factored form using these integers.

$(x + 1) ((x + 2) (x + 7))$

Step 2.2

Remove unnecessary parentheses.

$(x + 1) (x + 2) (x + 7)$