Precalculus Examples

$x^{4} + 15 x^{3} + 40 x^{2} - 39 x + 55 = 0$

Step 1

Factor the left side of the equation.

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Step 1.1

Factor $x^{4} + 15 x^{3} + 40 x^{2} - 39 x + 55$ using the rational roots test.

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Step 1.1.1

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.

$p = \pm 1, \pm 55, \pm 5, \pm 11$

$q = \pm 1$

Step 1.1.2

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

$\pm 1, \pm 55, \pm 5, \pm 11$

Step 1.1.3

Substitute $- 5$ and simplify the expression. In this case, the expression is equal to $0$ so $- 5$ is a root of the polynomial.

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Step 1.1.3.1

Substitute $- 5$ into the polynomial.

${(- 5)}^{4} + 15 {(- 5)}^{3} + 40 {(- 5)}^{2} - 39 \cdot - 5 + 55$

Step 1.1.3.2

Raise $- 5$ to the power of $4$ .

$625 + 15 {(- 5)}^{3} + 40 {(- 5)}^{2} - 39 \cdot - 5 + 55$

Step 1.1.3.3

Raise $- 5$ to the power of $3$ .

$625 + 15 \cdot - 125 + 40 {(- 5)}^{2} - 39 \cdot - 5 + 55$

Step 1.1.3.4

Multiply $15$ by $- 125$ .

$625 - 1875 + 40 {(- 5)}^{2} - 39 \cdot - 5 + 55$

Step 1.1.3.5

Subtract $1875$ from $625$ .

$- 1250 + 40 {(- 5)}^{2} - 39 \cdot - 5 + 55$

Step 1.1.3.6

Raise $- 5$ to the power of $2$ .

$- 1250 + 40 \cdot 25 - 39 \cdot - 5 + 55$

Step 1.1.3.7

Multiply $40$ by $25$ .

$- 1250 + 1000 - 39 \cdot - 5 + 55$

Step 1.1.3.8

Add $- 1250$ and $1000$ .

$- 250 - 39 \cdot - 5 + 55$

Step 1.1.3.9

Multiply $- 39$ by $- 5$ .

$- 250 + 195 + 55$

Step 1.1.3.10

Add $- 250$ and $195$ .

$- 55 + 55$

Step 1.1.3.11

Add $- 55$ and $55$ .

$0$

Step 1.1.4

Since $- 5$ is a known root, divide the polynomial by $x + 5$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{x^{4} + 15 x^{3} + 40 x^{2} - 39 x + 55}{x + 5}$

Step 1.1.5

Divide $x^{4} + 15 x^{3} + 40 x^{2} - 39 x + 55$ by $x + 5$ .

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Step 1.1.5.1

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

$x$

$5$

$x^{4}$

$15 x^{3}$

$40 x^{2}$

$39 x$

$55$

Step 1.1.5.2

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

$x^{3}$

$x$

$5$

$x^{4}$

$15 x^{3}$

$40 x^{2}$

$39 x$

$55$

Step 1.1.5.3

Multiply the new quotient term by the divisor.

$x^{3}$

$x$

$5$

$x^{4}$

$15 x^{3}$

$40 x^{2}$

$39 x$

$55$

$x^{4}$

$5 x^{3}$

Step 1.1.5.4

The expression needs to be subtracted from the dividend, so change all the signs in $x^{4} + 5 x^{3}$

$x^{3}$

$x$

$5$

$x^{4}$

$15 x^{3}$

$40 x^{2}$

$39 x$

$55$

$x^{4}$

$5 x^{3}$

Step 1.1.5.5

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

$x^{3}$

$x$

$5$

$x^{4}$

$15 x^{3}$

$40 x^{2}$

$39 x$

$55$

$x^{4}$

$5 x^{3}$

$10 x^{3}$

Step 1.1.5.6

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

$x^{3}$

$x$

$5$

$x^{4}$

$15 x^{3}$

$40 x^{2}$

$39 x$

$55$

$x^{4}$

$5 x^{3}$

$10 x^{3}$

$40 x^{2}$

Step 1.1.5.7

Divide the highest order term in the dividend $10 x^{3}$ by the highest order term in divisor $x$ .

$x^{3}$

$10 x^{2}$

$x$

$5$

$x^{4}$

$15 x^{3}$

$40 x^{2}$

$39 x$

$55$

$x^{4}$

$5 x^{3}$

$10 x^{3}$

$40 x^{2}$

Step 1.1.5.8

Multiply the new quotient term by the divisor.

$x^{3}$

$10 x^{2}$

$x$

$5$

$x^{4}$

$15 x^{3}$

$40 x^{2}$

$39 x$

$55$

$x^{4}$

$5 x^{3}$

$10 x^{3}$

$40 x^{2}$

$10 x^{3}$

$50 x^{2}$

Step 1.1.5.9

The expression needs to be subtracted from the dividend, so change all the signs in $10 x^{3} + 50 x^{2}$

$x^{3}$

$10 x^{2}$

$x$

$5$

$x^{4}$

$15 x^{3}$

$40 x^{2}$

$39 x$

$55$

$x^{4}$

$5 x^{3}$

$10 x^{3}$

$40 x^{2}$

$10 x^{3}$

$50 x^{2}$

Step 1.1.5.10

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

$x^{3}$

$10 x^{2}$

$x$

$5$

$x^{4}$

$15 x^{3}$

$40 x^{2}$

$39 x$

$55$

$x^{4}$

$5 x^{3}$

$10 x^{3}$

$40 x^{2}$

$10 x^{3}$

$50 x^{2}$

$10 x^{2}$

Step 1.1.5.11

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

$x^{3}$

$10 x^{2}$

$x$

$5$

$x^{4}$

$15 x^{3}$

$40 x^{2}$

$39 x$

$55$

$x^{4}$

$5 x^{3}$

$10 x^{3}$

$40 x^{2}$

$10 x^{3}$

$50 x^{2}$

$10 x^{2}$

$39 x$

Step 1.1.5.12

Divide the highest order term in the dividend $- 10 x^{2}$ by the highest order term in divisor $x$ .

$x^{3}$

$10 x^{2}$

$10 x$

$x$

$5$

$x^{4}$

$15 x^{3}$

$40 x^{2}$

$39 x$

$55$

$x^{4}$

$5 x^{3}$

$10 x^{3}$

$40 x^{2}$

$10 x^{3}$

$50 x^{2}$

$10 x^{2}$

$39 x$

Step 1.1.5.13

Multiply the new quotient term by the divisor.

$x^{3}$

$10 x^{2}$

$10 x$

$x$

$5$

$x^{4}$

$15 x^{3}$

$40 x^{2}$

$39 x$

$55$

$x^{4}$

$5 x^{3}$

$10 x^{3}$

$40 x^{2}$

$10 x^{3}$

$50 x^{2}$

$10 x^{2}$

$39 x$

$10 x^{2}$

$50 x$

Step 1.1.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $- 10 x^{2} - 50 x$

$x^{3}$

$10 x^{2}$

$10 x$

$x$

$5$

$x^{4}$

$15 x^{3}$

$40 x^{2}$

$39 x$

$55$

$x^{4}$

$5 x^{3}$

$10 x^{3}$

$40 x^{2}$

$10 x^{3}$

$50 x^{2}$

$10 x^{2}$

$39 x$

$10 x^{2}$

$50 x$

Step 1.1.5.15

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

$x^{3}$

$10 x^{2}$

$10 x$

$x$

$5$

$x^{4}$

$15 x^{3}$

$40 x^{2}$

$39 x$

$55$

$x^{4}$

$5 x^{3}$

$10 x^{3}$

$40 x^{2}$

$10 x^{3}$

$50 x^{2}$

$10 x^{2}$

$39 x$

$10 x^{2}$

$50 x$

$11 x$

Step 1.1.5.16

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

$x^{3}$

$10 x^{2}$

$10 x$

$x$

$5$

$x^{4}$

$15 x^{3}$

$40 x^{2}$

$39 x$

$55$

$x^{4}$

$5 x^{3}$

$10 x^{3}$

$40 x^{2}$

$10 x^{3}$

$50 x^{2}$

$10 x^{2}$

$39 x$

$10 x^{2}$

$50 x$

$11 x$

$55$

Step 1.1.5.17

Divide the highest order term in the dividend $11 x$ by the highest order term in divisor $x$ .

$x^{3}$

$10 x^{2}$

$10 x$

$11$

$x$

$5$

$x^{4}$

$15 x^{3}$

$40 x^{2}$

$39 x$

$55$

$x^{4}$

$5 x^{3}$

$10 x^{3}$

$40 x^{2}$

$10 x^{3}$

$50 x^{2}$

$10 x^{2}$

$39 x$

$10 x^{2}$

$50 x$

$11 x$

$55$

Step 1.1.5.18

Multiply the new quotient term by the divisor.

$x^{3}$

$10 x^{2}$

$10 x$

$11$

$x$

$5$

$x^{4}$

$15 x^{3}$

$40 x^{2}$

$39 x$

$55$

$x^{4}$

$5 x^{3}$

$10 x^{3}$

$40 x^{2}$

$10 x^{3}$

$50 x^{2}$

$10 x^{2}$

$39 x$

$10 x^{2}$

$50 x$

$11 x$

$55$

$11 x$

$55$

Step 1.1.5.19

The expression needs to be subtracted from the dividend, so change all the signs in $11 x + 55$

$x^{3}$

$10 x^{2}$

$10 x$

$11$

$x$

$5$

$x^{4}$

$15 x^{3}$

$40 x^{2}$

$39 x$

$55$

$x^{4}$

$5 x^{3}$

$10 x^{3}$

$40 x^{2}$

$10 x^{3}$

$50 x^{2}$

$10 x^{2}$

$39 x$

$10 x^{2}$

$50 x$

$11 x$

$55$

$11 x$

$55$

Step 1.1.5.20

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

$x^{3}$

$10 x^{2}$

$10 x$

$11$

$x$

$5$

$x^{4}$

$15 x^{3}$

$40 x^{2}$

$39 x$

$55$

$x^{4}$

$5 x^{3}$

$10 x^{3}$

$40 x^{2}$

$10 x^{3}$

$50 x^{2}$

$10 x^{2}$

$39 x$

$10 x^{2}$

$50 x$

$11 x$

$55$

$11 x$

$55$

$0$

Step 1.1.5.21

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

$x^{3} + 10 x^{2} - 10 x + 11$

Step 1.1.6

Write $x^{4} + 15 x^{3} + 40 x^{2} - 39 x + 55$ as a set of factors.

$(x + 5) (x^{3} + 10 x^{2} - 10 x + 11) = 0$

Step 1.2

Factor $x^{3} + 10 x^{2} - 10 x + 11$ using the rational roots test.

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Step 1.2.1

Factor $x^{3} + 10 x^{2} - 10 x + 11$ using the rational roots test.

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Step 1.2.1.1

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.

$p = \pm 1, \pm 11$

$q = \pm 1$

Step 1.2.1.2

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

$\pm 1, \pm 11$

Step 1.2.1.3

Substitute $- 11$ and simplify the expression. In this case, the expression is equal to $0$ so $- 11$ is a root of the polynomial.

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Step 1.2.1.3.1

Substitute $- 11$ into the polynomial.

${(- 11)}^{3} + 10 {(- 11)}^{2} - 10 \cdot - 11 + 11$

Step 1.2.1.3.2

Raise $- 11$ to the power of $3$ .

$- 1331 + 10 {(- 11)}^{2} - 10 \cdot - 11 + 11$

Step 1.2.1.3.3

Raise $- 11$ to the power of $2$ .

$- 1331 + 10 \cdot 121 - 10 \cdot - 11 + 11$

Step 1.2.1.3.4

Multiply $10$ by $121$ .

$- 1331 + 1210 - 10 \cdot - 11 + 11$

Step 1.2.1.3.5

Add $- 1331$ and $1210$ .

$- 121 - 10 \cdot - 11 + 11$

Step 1.2.1.3.6

Multiply $- 10$ by $- 11$ .

$- 121 + 110 + 11$

Step 1.2.1.3.7

Add $- 121$ and $110$ .

$- 11 + 11$

Step 1.2.1.3.8

Add $- 11$ and $11$ .

$0$

Step 1.2.1.4

Since $- 11$ is a known root, divide the polynomial by $x + 11$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{x^{3} + 10 x^{2} - 10 x + 11}{x + 11}$

Step 1.2.1.5

Divide $x^{3} + 10 x^{2} - 10 x + 11$ by $x + 11$ .

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Step 1.2.1.5.1

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

$x$

$11$

$x^{3}$

$10 x^{2}$

$10 x$

$11$

Step 1.2.1.5.2

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

					$x^{2}$
	$x$	+	$11$		$x^{3}$	+	$10 x^{2}$	-	$10 x$	+	$11$

Step 1.2.1.5.3

Multiply the new quotient term by the divisor.

				$x^{2}$
$x$	+	$11$		$x^{3}$	+	$10 x^{2}$	-	$10 x$	+	$11$
			+	$x^{3}$	+	$11 x^{2}$

Step 1.2.1.5.4

The expression needs to be subtracted from the dividend, so change all the signs in $x^{3} + 11 x^{2}$

				$x^{2}$
$x$	+	$11$		$x^{3}$	+	$10 x^{2}$	-	$10 x$	+	$11$
			-	$x^{3}$	-	$11 x^{2}$

Step 1.2.1.5.5

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

				$x^{2}$
$x$	+	$11$		$x^{3}$	+	$10 x^{2}$	-	$10 x$	+	$11$
			-	$x^{3}$	-	$11 x^{2}$
					-	$x^{2}$

Step 1.2.1.5.6

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

				$x^{2}$
$x$	+	$11$		$x^{3}$	+	$10 x^{2}$	-	$10 x$	+	$11$
			-	$x^{3}$	-	$11 x^{2}$
					-	$x^{2}$	-	$10 x$

Step 1.2.1.5.7

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

				$x^{2}$	-	$x$
$x$	+	$11$		$x^{3}$	+	$10 x^{2}$	-	$10 x$	+	$11$
			-	$x^{3}$	-	$11 x^{2}$
					-	$x^{2}$	-	$10 x$

Step 1.2.1.5.8

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$x$
$x$	+	$11$		$x^{3}$	+	$10 x^{2}$	-	$10 x$	+	$11$
			-	$x^{3}$	-	$11 x^{2}$
					-	$x^{2}$	-	$10 x$
					-	$x^{2}$	-	$11 x$

Step 1.2.1.5.9

The expression needs to be subtracted from the dividend, so change all the signs in $- x^{2} - 11 x$

				$x^{2}$	-	$x$
$x$	+	$11$		$x^{3}$	+	$10 x^{2}$	-	$10 x$	+	$11$
			-	$x^{3}$	-	$11 x^{2}$
					-	$x^{2}$	-	$10 x$
					+	$x^{2}$	+	$11 x$

Step 1.2.1.5.10

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

				$x^{2}$	-	$x$
$x$	+	$11$		$x^{3}$	+	$10 x^{2}$	-	$10 x$	+	$11$
			-	$x^{3}$	-	$11 x^{2}$
					-	$x^{2}$	-	$10 x$
					+	$x^{2}$	+	$11 x$
							+	$x$

Step 1.2.1.5.11

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

				$x^{2}$	-	$x$
$x$	+	$11$		$x^{3}$	+	$10 x^{2}$	-	$10 x$	+	$11$
			-	$x^{3}$	-	$11 x^{2}$
					-	$x^{2}$	-	$10 x$
					+	$x^{2}$	+	$11 x$
							+	$x$	+	$11$

Step 1.2.1.5.12

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

				$x^{2}$	-	$x$	+	$1$
$x$	+	$11$		$x^{3}$	+	$10 x^{2}$	-	$10 x$	+	$11$
			-	$x^{3}$	-	$11 x^{2}$
					-	$x^{2}$	-	$10 x$
					+	$x^{2}$	+	$11 x$
							+	$x$	+	$11$

Step 1.2.1.5.13

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$x$	+	$1$
$x$	+	$11$		$x^{3}$	+	$10 x^{2}$	-	$10 x$	+	$11$
			-	$x^{3}$	-	$11 x^{2}$
					-	$x^{2}$	-	$10 x$
					+	$x^{2}$	+	$11 x$
							+	$x$	+	$11$
							+	$x$	+	$11$

Step 1.2.1.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $x + 11$

				$x^{2}$	-	$x$	+	$1$
$x$	+	$11$		$x^{3}$	+	$10 x^{2}$	-	$10 x$	+	$11$
			-	$x^{3}$	-	$11 x^{2}$
					-	$x^{2}$	-	$10 x$
					+	$x^{2}$	+	$11 x$
							+	$x$	+	$11$
							-	$x$	-	$11$

Step 1.2.1.5.15

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

				$x^{2}$	-	$x$	+	$1$
$x$	+	$11$		$x^{3}$	+	$10 x^{2}$	-	$10 x$	+	$11$
			-	$x^{3}$	-	$11 x^{2}$
					-	$x^{2}$	-	$10 x$
					+	$x^{2}$	+	$11 x$
							+	$x$	+	$11$
							-	$x$	-	$11$
										$0$

Step 1.2.1.5.16

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

$x^{2} - x + 1$

Step 1.2.1.6

Write $x^{3} + 10 x^{2} - 10 x + 11$ as a set of factors.

$(x + 5) ((x + 11) (x^{2} - x + 1)) = 0$

Step 1.2.2

Remove unnecessary parentheses.

$(x + 5) (x + 11) (x^{2} - x + 1) = 0$

Step 2

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

$x + 5 = 0$

$x + 11 = 0$

$x^{2} - x + 1 = 0$

Step 3

Set $x + 5$ equal to $0$ and solve for $x$ .

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Step 3.1

Set $x + 5$ equal to $0$ .

$x + 5 = 0$

Step 3.2

Subtract $5$ from both sides of the equation.

$x = - 5$

Step 4

Set $x + 11$ equal to $0$ and solve for $x$ .

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Step 4.1

Set $x + 11$ equal to $0$ .

$x + 11 = 0$

Step 4.2

Subtract $11$ from both sides of the equation.

$x = - 11$

Step 5

Set $x^{2} - x + 1$ equal to $0$ and solve for $x$ .

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Step 5.1

Set $x^{2} - x + 1$ equal to $0$ .

$x^{2} - x + 1 = 0$

Step 5.2

Solve $x^{2} - x + 1 = 0$ for $x$ .

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Step 5.2.1

Use the quadratic formula to find the solutions.

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

Step 5.2.2

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

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

Step 5.2.3

Simplify.

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Step 5.2.3.1

Simplify the numerator.

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Step 5.2.3.1.1

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

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

Step 5.2.3.1.2

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

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Step 5.2.3.1.2.1

Multiply $- 4$ by $1$ .

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

Step 5.2.3.1.2.2

Multiply $- 4$ by $1$ .

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

Step 5.2.3.1.3

Subtract $4$ from $1$ .

$x = \frac{1 \pm \sqrt{- 3}}{2 \cdot 1}$

Step 5.2.3.1.4

Rewrite $- 3$ as $- 1 (3)$ .

$x = \frac{1 \pm \sqrt{- 1 \cdot 3}}{2 \cdot 1}$

Step 5.2.3.1.5

Rewrite $\sqrt{- 1 (3)}$ as $\sqrt{- 1} \cdot \sqrt{3}$ .

$x = \frac{1 \pm \sqrt{- 1} \cdot \sqrt{3}}{2 \cdot 1}$

Step 5.2.3.1.6

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

$x = \frac{1 \pm i \sqrt{3}}{2 \cdot 1}$

Step 5.2.3.2

Multiply $2$ by $1$ .

$x = \frac{1 \pm i \sqrt{3}}{2}$

Step 5.2.4

The final answer is the combination of both solutions.

$x = \frac{1 + i \sqrt{3}}{2}, \frac{1 - i \sqrt{3}}{2}$

Step 6

The final solution is all the values that make $(x + 5) (x + 11) (x^{2} - x + 1) = 0$ true.

$x = - 5, - 11, \frac{1 + i \sqrt{3}}{2}, \frac{1 - i \sqrt{3}}{2}$