Precalculus Examples

$x^{4} + 6 x^{3} + 4 x^{2} - 3 x + 10 = 0$

Step 1

Factor the left side of the equation.

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Step 1.1

Factor $x^{4} + 6 x^{3} + 4 x^{2} - 3 x + 10$ 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.

$p = \pm 1, \pm 10, \pm 2, \pm 5$

$q = \pm 1$

Step 1.1.2

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

$\pm 1, \pm 10, \pm 2, \pm 5$

Step 1.1.3

Substitute $- 2$ and simplify the expression. In this case, the expression is equal to $0$ so $- 2$ is a root of the polynomial.

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Step 1.1.3.1

Substitute $- 2$ into the polynomial.

${(- 2)}^{4} + 6 {(- 2)}^{3} + 4 {(- 2)}^{2} - 3 \cdot - 2 + 10$

Step 1.1.3.2

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

$16 + 6 {(- 2)}^{3} + 4 {(- 2)}^{2} - 3 \cdot - 2 + 10$

Step 1.1.3.3

Raise $- 2$ to the power of $3$ .

$16 + 6 \cdot - 8 + 4 {(- 2)}^{2} - 3 \cdot - 2 + 10$

Step 1.1.3.4

Multiply $6$ by $- 8$ .

$16 - 48 + 4 {(- 2)}^{2} - 3 \cdot - 2 + 10$

Step 1.1.3.5

Subtract $48$ from $16$ .

$- 32 + 4 {(- 2)}^{2} - 3 \cdot - 2 + 10$

Step 1.1.3.6

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

$- 32 + 4 \cdot 4 - 3 \cdot - 2 + 10$

Step 1.1.3.7

Multiply $4$ by $4$ .

$- 32 + 16 - 3 \cdot - 2 + 10$

Step 1.1.3.8

Add $- 32$ and $16$ .

$- 16 - 3 \cdot - 2 + 10$

Step 1.1.3.9

Multiply $- 3$ by $- 2$ .

$- 16 + 6 + 10$

Step 1.1.3.10

Add $- 16$ and $6$ .

$- 10 + 10$

Step 1.1.3.11

Add $- 10$ and $10$ .

$0$

Step 1.1.4

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.

$\frac{x^{4} + 6 x^{3} + 4 x^{2} - 3 x + 10}{x + 2}$

Step 1.1.5

Divide $x^{4} + 6 x^{3} + 4 x^{2} - 3 x + 10$ by $x + 2$ .

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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 $0$ .

$x$

$2$

$x^{4}$

$6 x^{3}$

$4 x^{2}$

$3 x$

$10$

Step 1.1.5.2

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

$x^{3}$

$x$

$2$

$x^{4}$

$6 x^{3}$

$4 x^{2}$

$3 x$

$10$

Step 1.1.5.3

Multiply the new quotient term by the divisor.

$x^{3}$

$x$

$2$

$x^{4}$

$6 x^{3}$

$4 x^{2}$

$3 x$

$10$

$x^{4}$

$2 x^{3}$

Step 1.1.5.4

The expression needs to be subtracted from the dividend, so change all the signs in $x^{4} + 2 x^{3}$

$x^{3}$

$x$

$2$

$x^{4}$

$6 x^{3}$

$4 x^{2}$

$3 x$

$10$

$x^{4}$

$2 x^{3}$

Step 1.1.5.5

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

$x^{3}$

$x$

$2$

$x^{4}$

$6 x^{3}$

$4 x^{2}$

$3 x$

$10$

$x^{4}$

$2 x^{3}$

$4 x^{3}$

Step 1.1.5.6

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

$x^{3}$

$x$

$2$

$x^{4}$

$6 x^{3}$

$4 x^{2}$

$3 x$

$10$

$x^{4}$

$2 x^{3}$

$4 x^{3}$

$4 x^{2}$

Step 1.1.5.7

Divide the highest order term in the dividend $4 x^{3}$ by the highest order term in divisor $x$ .

$x^{3}$

$4 x^{2}$

$x$

$2$

$x^{4}$

$6 x^{3}$

$4 x^{2}$

$3 x$

$10$

$x^{4}$

$2 x^{3}$

$4 x^{3}$

$4 x^{2}$

Step 1.1.5.8

Multiply the new quotient term by the divisor.

$x^{3}$

$4 x^{2}$

$x$

$2$

$x^{4}$

$6 x^{3}$

$4 x^{2}$

$3 x$

$10$

$x^{4}$

$2 x^{3}$

$4 x^{3}$

$4 x^{2}$

$4 x^{3}$

$8 x^{2}$

Step 1.1.5.9

The expression needs to be subtracted from the dividend, so change all the signs in $4 x^{3} + 8 x^{2}$

$x^{3}$

$4 x^{2}$

$x$

$2$

$x^{4}$

$6 x^{3}$

$4 x^{2}$

$3 x$

$10$

$x^{4}$

$2 x^{3}$

$4 x^{3}$

$4 x^{2}$

$4 x^{3}$

$8 x^{2}$

Step 1.1.5.10

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

$x^{3}$

$4 x^{2}$

$x$

$2$

$x^{4}$

$6 x^{3}$

$4 x^{2}$

$3 x$

$10$

$x^{4}$

$2 x^{3}$

$4 x^{3}$

$4 x^{2}$

$4 x^{3}$

$8 x^{2}$

$4 x^{2}$

Step 1.1.5.11

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

$x^{3}$

$4 x^{2}$

$x$

$2$

$x^{4}$

$6 x^{3}$

$4 x^{2}$

$3 x$

$10$

$x^{4}$

$2 x^{3}$

$4 x^{3}$

$4 x^{2}$

$4 x^{3}$

$8 x^{2}$

$4 x^{2}$

$3 x$

Step 1.1.5.12

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

$x^{3}$

$4 x^{2}$

$4 x$

$x$

$2$

$x^{4}$

$6 x^{3}$

$4 x^{2}$

$3 x$

$10$

$x^{4}$

$2 x^{3}$

$4 x^{3}$

$4 x^{2}$

$4 x^{3}$

$8 x^{2}$

$4 x^{2}$

$3 x$

Step 1.1.5.13

Multiply the new quotient term by the divisor.

$x^{3}$

$4 x^{2}$

$4 x$

$x$

$2$

$x^{4}$

$6 x^{3}$

$4 x^{2}$

$3 x$

$10$

$x^{4}$

$2 x^{3}$

$4 x^{3}$

$4 x^{2}$

$4 x^{3}$

$8 x^{2}$

$4 x^{2}$

$3 x$

$4 x^{2}$

$8 x$

Step 1.1.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $- 4 x^{2} - 8 x$

$x^{3}$

$4 x^{2}$

$4 x$

$x$

$2$

$x^{4}$

$6 x^{3}$

$4 x^{2}$

$3 x$

$10$

$x^{4}$

$2 x^{3}$

$4 x^{3}$

$4 x^{2}$

$4 x^{3}$

$8 x^{2}$

$4 x^{2}$

$3 x$

$4 x^{2}$

$8 x$

Step 1.1.5.15

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

$x^{3}$

$4 x^{2}$

$4 x$

$x$

$2$

$x^{4}$

$6 x^{3}$

$4 x^{2}$

$3 x$

$10$

$x^{4}$

$2 x^{3}$

$4 x^{3}$

$4 x^{2}$

$4 x^{3}$

$8 x^{2}$

$4 x^{2}$

$3 x$

$4 x^{2}$

$8 x$

$5 x$

Step 1.1.5.16

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

$x^{3}$

$4 x^{2}$

$4 x$

$x$

$2$

$x^{4}$

$6 x^{3}$

$4 x^{2}$

$3 x$

$10$

$x^{4}$

$2 x^{3}$

$4 x^{3}$

$4 x^{2}$

$4 x^{3}$

$8 x^{2}$

$4 x^{2}$

$3 x$

$4 x^{2}$

$8 x$

$5 x$

$10$

Step 1.1.5.17

Divide the highest order term in the dividend $5 x$ by the highest order term in divisor $x$ .

$x^{3}$

$4 x^{2}$

$4 x$

$5$

$x$

$2$

$x^{4}$

$6 x^{3}$

$4 x^{2}$

$3 x$

$10$

$x^{4}$

$2 x^{3}$

$4 x^{3}$

$4 x^{2}$

$4 x^{3}$

$8 x^{2}$

$4 x^{2}$

$3 x$

$4 x^{2}$

$8 x$

$5 x$

$10$

Step 1.1.5.18

Multiply the new quotient term by the divisor.

$x^{3}$

$4 x^{2}$

$4 x$

$5$

$x$

$2$

$x^{4}$

$6 x^{3}$

$4 x^{2}$

$3 x$

$10$

$x^{4}$

$2 x^{3}$

$4 x^{3}$

$4 x^{2}$

$4 x^{3}$

$8 x^{2}$

$4 x^{2}$

$3 x$

$4 x^{2}$

$8 x$

$5 x$

$10$

$5 x$

$10$

Step 1.1.5.19

The expression needs to be subtracted from the dividend, so change all the signs in $5 x + 10$

$x^{3}$

$4 x^{2}$

$4 x$

$5$

$x$

$2$

$x^{4}$

$6 x^{3}$

$4 x^{2}$

$3 x$

$10$

$x^{4}$

$2 x^{3}$

$4 x^{3}$

$4 x^{2}$

$4 x^{3}$

$8 x^{2}$

$4 x^{2}$

$3 x$

$4 x^{2}$

$8 x$

$5 x$

$10$

$5 x$

$10$

Step 1.1.5.20

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

$x^{3}$

$4 x^{2}$

$4 x$

$5$

$x$

$2$

$x^{4}$

$6 x^{3}$

$4 x^{2}$

$3 x$

$10$

$x^{4}$

$2 x^{3}$

$4 x^{3}$

$4 x^{2}$

$4 x^{3}$

$8 x^{2}$

$4 x^{2}$

$3 x$

$4 x^{2}$

$8 x$

$5 x$

$10$

$5 x$

$10$

$0$

Step 1.1.5.21

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

$x^{3} + 4 x^{2} - 4 x + 5$

Step 1.1.6

Write $x^{4} + 6 x^{3} + 4 x^{2} - 3 x + 10$ as a set of factors.

$(x + 2) (x^{3} + 4 x^{2} - 4 x + 5) = 0$

Step 1.2

Factor $x^{3} + 4 x^{2} - 4 x + 5$ using the rational roots test.

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Step 1.2.1

Factor $x^{3} + 4 x^{2} - 4 x + 5$ 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.

$p = \pm 1, \pm 5$

$q = \pm 1$

Step 1.2.1.2

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

$\pm 1, \pm 5$

Step 1.2.1.3

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 1.2.1.3.1

Substitute $- 5$ into the polynomial.

${(- 5)}^{3} + 4 {(- 5)}^{2} - 4 \cdot - 5 + 5$

Step 1.2.1.3.2

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

$- 125 + 4 {(- 5)}^{2} - 4 \cdot - 5 + 5$

Step 1.2.1.3.3

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

$- 125 + 4 \cdot 25 - 4 \cdot - 5 + 5$

Step 1.2.1.3.4

Multiply $4$ by $25$ .

$- 125 + 100 - 4 \cdot - 5 + 5$

Step 1.2.1.3.5

Add $- 125$ and $100$ .

$- 25 - 4 \cdot - 5 + 5$

Step 1.2.1.3.6

Multiply $- 4$ by $- 5$ .

$- 25 + 20 + 5$

Step 1.2.1.3.7

Add $- 25$ and $20$ .

$- 5 + 5$

Step 1.2.1.3.8

Add $- 5$ and $5$ .

$0$

Step 1.2.1.4

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.

$\frac{x^{3} + 4 x^{2} - 4 x + 5}{x + 5}$

Step 1.2.1.5

Divide $x^{3} + 4 x^{2} - 4 x + 5$ by $x + 5$ .

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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 $0$ .

$x$

$5$

$x^{3}$

$4 x^{2}$

$4 x$

$5$

Step 1.2.1.5.2

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

					$x^{2}$
	$x$	+	$5$		$x^{3}$	+	$4 x^{2}$	-	$4 x$	+	$5$

Step 1.2.1.5.3

Multiply the new quotient term by the divisor.

				$x^{2}$
$x$	+	$5$		$x^{3}$	+	$4 x^{2}$	-	$4 x$	+	$5$
			+	$x^{3}$	+	$5 x^{2}$

Step 1.2.1.5.4

The expression needs to be subtracted from the dividend, so change all the signs in $x^{3} + 5 x^{2}$

				$x^{2}$
$x$	+	$5$		$x^{3}$	+	$4 x^{2}$	-	$4 x$	+	$5$
			-	$x^{3}$	-	$5 x^{2}$

Step 1.2.1.5.5

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

				$x^{2}$
$x$	+	$5$		$x^{3}$	+	$4 x^{2}$	-	$4 x$	+	$5$
			-	$x^{3}$	-	$5 x^{2}$
					-	$x^{2}$

Step 1.2.1.5.6

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

				$x^{2}$
$x$	+	$5$		$x^{3}$	+	$4 x^{2}$	-	$4 x$	+	$5$
			-	$x^{3}$	-	$5 x^{2}$
					-	$x^{2}$	-	$4 x$

Step 1.2.1.5.7

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

				$x^{2}$	-	$x$
$x$	+	$5$		$x^{3}$	+	$4 x^{2}$	-	$4 x$	+	$5$
			-	$x^{3}$	-	$5 x^{2}$
					-	$x^{2}$	-	$4 x$

Step 1.2.1.5.8

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$x$
$x$	+	$5$		$x^{3}$	+	$4 x^{2}$	-	$4 x$	+	$5$
			-	$x^{3}$	-	$5 x^{2}$
					-	$x^{2}$	-	$4 x$
					-	$x^{2}$	-	$5 x$

Step 1.2.1.5.9

The expression needs to be subtracted from the dividend, so change all the signs in $- x^{2} - 5 x$

				$x^{2}$	-	$x$
$x$	+	$5$		$x^{3}$	+	$4 x^{2}$	-	$4 x$	+	$5$
			-	$x^{3}$	-	$5 x^{2}$
					-	$x^{2}$	-	$4 x$
					+	$x^{2}$	+	$5 x$

Step 1.2.1.5.10

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

				$x^{2}$	-	$x$
$x$	+	$5$		$x^{3}$	+	$4 x^{2}$	-	$4 x$	+	$5$
			-	$x^{3}$	-	$5 x^{2}$
					-	$x^{2}$	-	$4 x$
					+	$x^{2}$	+	$5 x$
							+	$x$

Step 1.2.1.5.11

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

				$x^{2}$	-	$x$
$x$	+	$5$		$x^{3}$	+	$4 x^{2}$	-	$4 x$	+	$5$
			-	$x^{3}$	-	$5 x^{2}$
					-	$x^{2}$	-	$4 x$
					+	$x^{2}$	+	$5 x$
							+	$x$	+	$5$

Step 1.2.1.5.12

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

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

Step 1.2.1.5.13

Multiply the new quotient term by the divisor.

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

Step 1.2.1.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $x + 5$

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

Step 1.2.1.5.15

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

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

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 $x^{3} + 4 x^{2} - 4 x + 5$ as a set of factors.

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

Step 1.2.2

Remove unnecessary parentheses.

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

Step 2

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

$x + 2 = 0$

$x + 5 = 0$

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

Step 3

Set $x + 2$ equal to $0$ and solve for $x$ .

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Step 3.1

Set $x + 2$ equal to $0$ .

$x + 2 = 0$

Step 3.2

Subtract $2$ from both sides of the equation.

$x = - 2$

Step 4

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

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Step 4.1

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

$x + 5 = 0$

Step 4.2

Subtract $5$ from both sides of the equation.

$x = - 5$

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

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