Precalculus Examples

$x^{3} - 5 x^{2} + 17 x - 13 = 0$

Step 1

Factor $x^{3} - 5 x^{2} + 17 x - 13$ 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.

$p = \pm 1, \pm 13$

$q = \pm 1$

Step 1.2

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

$\pm 1, \pm 13$

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.

$1^{3} - 5 \cdot 1^{2} + 17 \cdot 1 - 13$

Step 1.3.2

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

$1 - 5 \cdot 1^{2} + 17 \cdot 1 - 13$

Step 1.3.3

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

$1 - 5 \cdot 1 + 17 \cdot 1 - 13$

Step 1.3.4

Multiply $- 5$ by $1$ .

$1 - 5 + 17 \cdot 1 - 13$

Step 1.3.5

Subtract $5$ from $1$ .

$- 4 + 17 \cdot 1 - 13$

Step 1.3.6

Multiply $17$ by $1$ .

$- 4 + 17 - 13$

Step 1.3.7

Add $- 4$ and $17$ .

$13 - 13$

Step 1.3.8

Subtract $13$ from $13$ .

$0$

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.

$\frac{x^{3} - 5 x^{2} + 17 x - 13}{x - 1}$

Step 1.5

Divide $x^{3} - 5 x^{2} + 17 x - 13$ 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 $0$ .

$x$

$1$

$x^{3}$

$5 x^{2}$

$17 x$

$13$

Step 1.5.2

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

					$x^{2}$
	$x$	-	$1$		$x^{3}$	-	$5 x^{2}$	+	$17 x$	-	$13$

Step 1.5.3

Multiply the new quotient term by the divisor.

				$x^{2}$
$x$	-	$1$		$x^{3}$	-	$5 x^{2}$	+	$17 x$	-	$13$
			+	$x^{3}$	-	$x^{2}$

Step 1.5.4

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

				$x^{2}$
$x$	-	$1$		$x^{3}$	-	$5 x^{2}$	+	$17 x$	-	$13$
			-	$x^{3}$	+	$x^{2}$

Step 1.5.5

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

				$x^{2}$
$x$	-	$1$		$x^{3}$	-	$5 x^{2}$	+	$17 x$	-	$13$
			-	$x^{3}$	+	$x^{2}$
					-	$4 x^{2}$

Step 1.5.6

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

				$x^{2}$
$x$	-	$1$		$x^{3}$	-	$5 x^{2}$	+	$17 x$	-	$13$
			-	$x^{3}$	+	$x^{2}$
					-	$4 x^{2}$	+	$17 x$

Step 1.5.7

Divide the highest order term in the dividend $- 4 x^{2}$ by the highest order term in divisor $x$ .

				$x^{2}$	-	$4 x$
$x$	-	$1$		$x^{3}$	-	$5 x^{2}$	+	$17 x$	-	$13$
			-	$x^{3}$	+	$x^{2}$
					-	$4 x^{2}$	+	$17 x$

Step 1.5.8

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$4 x$
$x$	-	$1$		$x^{3}$	-	$5 x^{2}$	+	$17 x$	-	$13$
			-	$x^{3}$	+	$x^{2}$
					-	$4 x^{2}$	+	$17 x$
					-	$4 x^{2}$	+	$4 x$

Step 1.5.9

The expression needs to be subtracted from the dividend, so change all the signs in $- 4 x^{2} + 4 x$

				$x^{2}$	-	$4 x$
$x$	-	$1$		$x^{3}$	-	$5 x^{2}$	+	$17 x$	-	$13$
			-	$x^{3}$	+	$x^{2}$
					-	$4 x^{2}$	+	$17 x$
					+	$4 x^{2}$	-	$4 x$

Step 1.5.10

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

				$x^{2}$	-	$4 x$
$x$	-	$1$		$x^{3}$	-	$5 x^{2}$	+	$17 x$	-	$13$
			-	$x^{3}$	+	$x^{2}$
					-	$4 x^{2}$	+	$17 x$
					+	$4 x^{2}$	-	$4 x$
							+	$13 x$

Step 1.5.11

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

				$x^{2}$	-	$4 x$
$x$	-	$1$		$x^{3}$	-	$5 x^{2}$	+	$17 x$	-	$13$
			-	$x^{3}$	+	$x^{2}$
					-	$4 x^{2}$	+	$17 x$
					+	$4 x^{2}$	-	$4 x$
							+	$13 x$	-	$13$

Step 1.5.12

Divide the highest order term in the dividend $13 x$ by the highest order term in divisor $x$ .

				$x^{2}$	-	$4 x$	+	$13$
$x$	-	$1$		$x^{3}$	-	$5 x^{2}$	+	$17 x$	-	$13$
			-	$x^{3}$	+	$x^{2}$
					-	$4 x^{2}$	+	$17 x$
					+	$4 x^{2}$	-	$4 x$
							+	$13 x$	-	$13$

Step 1.5.13

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$4 x$	+	$13$
$x$	-	$1$		$x^{3}$	-	$5 x^{2}$	+	$17 x$	-	$13$
			-	$x^{3}$	+	$x^{2}$
					-	$4 x^{2}$	+	$17 x$
					+	$4 x^{2}$	-	$4 x$
							+	$13 x$	-	$13$
							+	$13 x$	-	$13$

Step 1.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $13 x - 13$

				$x^{2}$	-	$4 x$	+	$13$
$x$	-	$1$		$x^{3}$	-	$5 x^{2}$	+	$17 x$	-	$13$
			-	$x^{3}$	+	$x^{2}$
					-	$4 x^{2}$	+	$17 x$
					+	$4 x^{2}$	-	$4 x$
							+	$13 x$	-	$13$
							-	$13 x$	+	$13$

Step 1.5.15

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

				$x^{2}$	-	$4 x$	+	$13$
$x$	-	$1$		$x^{3}$	-	$5 x^{2}$	+	$17 x$	-	$13$
			-	$x^{3}$	+	$x^{2}$
					-	$4 x^{2}$	+	$17 x$
					+	$4 x^{2}$	-	$4 x$
							+	$13 x$	-	$13$
							-	$13 x$	+	$13$
										$0$

Step 1.5.16

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

$x^{2} - 4 x + 13$

Step 1.6

Write $x^{3} - 5 x^{2} + 17 x - 13$ as a set of factors.

$(x - 1) (x^{2} - 4 x + 13) = 0$

Step 2

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x - 1 = 0$

$x^{2} - 4 x + 13 = 0$

Step 3

Set $x - 1$ equal to $0$ and solve for $x$ .

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Step 3.1

Set $x - 1$ equal to $0$ .

$x - 1 = 0$

Step 3.2

Add $1$ to both sides of the equation.

$x = 1$

Step 4

Set $x^{2} - 4 x + 13$ equal to $0$ and solve for $x$ .

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Step 4.1

Set $x^{2} - 4 x + 13$ equal to $0$ .

$x^{2} - 4 x + 13 = 0$

Step 4.2

Solve $x^{2} - 4 x + 13 = 0$ for $x$ .

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Step 4.2.1

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 4.2.2

Substitute the values $a = 1$ , $b = - 4$ , and $c = 13$ into the quadratic formula and solve for $x$ .

$\frac{4 \pm \sqrt{{(- 4)}^{2} - 4 \cdot (1 \cdot 13)}}{2 \cdot 1}$

Step 4.2.3

Simplify.

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Step 4.2.3.1

Simplify the numerator.

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Step 4.2.3.1.1

Raise $- 4$ to the power of $2$ .

$x = \frac{4 \pm \sqrt{16 - 4 \cdot 1 \cdot 13}}{2 \cdot 1}$

Step 4.2.3.1.2

Multiply $- 4 \cdot 1 \cdot 13$ .

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Step 4.2.3.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{4 \pm \sqrt{16 - 4 \cdot 13}}{2 \cdot 1}$

Step 4.2.3.1.2.2

Multiply $- 4$ by $13$ .

$x = \frac{4 \pm \sqrt{16 - 52}}{2 \cdot 1}$

Step 4.2.3.1.3

Subtract $52$ from $16$ .

$x = \frac{4 \pm \sqrt{- 36}}{2 \cdot 1}$

Step 4.2.3.1.4

Rewrite $- 36$ as $- 1 (36)$ .

$x = \frac{4 \pm \sqrt{- 1 \cdot 36}}{2 \cdot 1}$

Step 4.2.3.1.5

Rewrite $\sqrt{- 1 (36)}$ as $\sqrt{- 1} \cdot \sqrt{36}$ .

$x = \frac{4 \pm \sqrt{- 1} \cdot \sqrt{36}}{2 \cdot 1}$

Step 4.2.3.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{4 \pm i \cdot \sqrt{36}}{2 \cdot 1}$

Step 4.2.3.1.7

Rewrite $36$ as $6^{2}$ .

$x = \frac{4 \pm i \cdot \sqrt{6^{2}}}{2 \cdot 1}$

Step 4.2.3.1.8

Pull terms out from under the radical, assuming positive real numbers.

$x = \frac{4 \pm i \cdot 6}{2 \cdot 1}$

Step 4.2.3.1.9

Move $6$ to the left of $i$ .

$x = \frac{4 \pm 6 i}{2 \cdot 1}$

Step 4.2.3.2

Multiply $2$ by $1$ .

$x = \frac{4 \pm 6 i}{2}$

Step 4.2.3.3

Simplify $\frac{4 \pm 6 i}{2}$ .

$x = 2 \pm 3 i$

Step 4.2.4

The final answer is the combination of both solutions.

$x = 2 + 3 i, 2 - 3 i$

Step 5

The final solution is all the values that make $(x - 1) (x^{2} - 4 x + 13) = 0$ true.

$x = 1, 2 + 3 i, 2 - 3 i$