Precalculus Examples

$x^{4} + 2 x^{3} - 5 x^{2} - 8 x + 4 = 0$

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.

$p = \pm 1, \pm 2, \pm 4$

$q = \pm 1$

Step 2

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

$\pm 1, \pm 2, \pm 4$

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.

${(2)}^{4} + 2 {(2)}^{3} - 5 {(2)}^{2} - 8 \cdot 2 + 4$

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 $4$ .

$16 + 2 {(2)}^{3} - 5 {(2)}^{2} - 8 \cdot 2 + 4$

Step 4.1.2

Multiply $2$ by ${(2)}^{3}$ by adding the exponents.

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Step 4.1.2.1

Multiply $2$ by ${(2)}^{3}$ .

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Step 4.1.2.1.1

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

$16 + 2^{1} {(2)}^{3} - 5 {(2)}^{2} - 8 \cdot 2 + 4$

Step 4.1.2.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$16 + 2^{1 + 3} - 5 {(2)}^{2} - 8 \cdot 2 + 4$

Step 4.1.2.2

Add $1$ and $3$ .

$16 + 2^{4} - 5 {(2)}^{2} - 8 \cdot 2 + 4$

Step 4.1.3

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

$16 + 16 - 5 {(2)}^{2} - 8 \cdot 2 + 4$

Step 4.1.4

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

$16 + 16 - 5 \cdot 4 - 8 \cdot 2 + 4$

Step 4.1.5

Multiply $- 5$ by $4$ .

$16 + 16 - 20 - 8 \cdot 2 + 4$

Step 4.1.6

Multiply $- 8$ by $2$ .

$16 + 16 - 20 - 16 + 4$

Step 4.2

Simplify by adding and subtracting.

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Step 4.2.1

Add $16$ and $16$ .

$32 - 20 - 16 + 4$

Step 4.2.2

Subtract $20$ from $32$ .

$12 - 16 + 4$

Step 4.2.3

Subtract $16$ from $12$ .

$- 4 + 4$

Step 4.2.4

Add $- 4$ and $4$ .

$0$

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.

$\frac{x^{4} + 2 x^{3} - 5 x^{2} - 8 x + 4}{x - 2}$

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.

$2$	$1$	$2$	$- 5$	$- 8$	$4$

Step 6.2

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

$2$	$1$	$2$	$- 5$	$- 8$	$4$

	$1$

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 $(2)$ .

$2$	$1$	$2$	$- 5$	$- 8$	$4$
		$2$
	$1$

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.

$2$	$1$	$2$	$- 5$	$- 8$	$4$
		$2$
	$1$	$4$

Step 6.5

Multiply the newest entry in the result $(4)$ by the divisor $(2)$ and place the result of $(8)$ under the next term in the dividend $(- 5)$ .

$2$	$1$	$2$	$- 5$	$- 8$	$4$
		$2$	$8$
	$1$	$4$

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.

$2$	$1$	$2$	$- 5$	$- 8$	$4$
		$2$	$8$
	$1$	$4$	$3$

Step 6.7

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 $(- 8)$ .

$2$	$1$	$2$	$- 5$	$- 8$	$4$
		$2$	$8$	$6$
	$1$	$4$	$3$

Step 6.8

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$2$	$1$	$2$	$- 5$	$- 8$	$4$
		$2$	$8$	$6$
	$1$	$4$	$3$	$- 2$

Step 6.9

Multiply the newest entry in the result $(- 2)$ by the divisor $(2)$ and place the result of $(- 4)$ under the next term in the dividend $(4)$ .

$2$	$1$	$2$	$- 5$	$- 8$	$4$
		$2$	$8$	$6$	$- 4$
	$1$	$4$	$3$	$- 2$

Step 6.10

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$2$	$1$	$2$	$- 5$	$- 8$	$4$
		$2$	$8$	$6$	$- 4$
	$1$	$4$	$3$	$- 2$	$0$

Step 6.11

All numbers except the last become the coefficients of the quotient polynomial. The last value in the result line is the remainder.

$1 x^{3} + 4 x^{2} + (3) x - 2$

Step 6.12

Simplify the quotient polynomial.

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

Step 7

Solve the equation to find any remaining roots.

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Step 7.1

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

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Step 7.1.1

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.

$p = \pm 1, \pm 2$

$q = \pm 1$

Step 7.1.2

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

$\pm 1, \pm 2$

Step 7.1.3

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 7.1.3.1

Substitute $- 2$ into the polynomial.

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

Step 7.1.3.2

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

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

Step 7.1.3.3

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

$- 8 + 4 \cdot 4 + 3 \cdot - 2 - 2$

Step 7.1.3.4

Multiply $4$ by $4$ .

$- 8 + 16 + 3 \cdot - 2 - 2$

Step 7.1.3.5

Add $- 8$ and $16$ .

$8 + 3 \cdot - 2 - 2$

Step 7.1.3.6

Multiply $3$ by $- 2$ .

$8 - 6 - 2$

Step 7.1.3.7

Subtract $6$ from $8$ .

$2 - 2$

Step 7.1.3.8

Subtract $2$ from $2$ .

$0$

Step 7.1.4

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.

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

Step 7.1.5

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

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Step 7.1.5.1

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

$x$

$2$

$x^{3}$

$4 x^{2}$

$3 x$

$2$

Step 7.1.5.2

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

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

Step 7.1.5.3

Multiply the new quotient term by the divisor.

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

Step 7.1.5.4

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

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

Step 7.1.5.5

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

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

Step 7.1.5.6

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

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

Step 7.1.5.7

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

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

Step 7.1.5.8

Multiply the new quotient term by the divisor.

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

Step 7.1.5.9

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

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

Step 7.1.5.10

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

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

Step 7.1.5.11

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

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

Step 7.1.5.12

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

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

Step 7.1.5.13

Multiply the new quotient term by the divisor.

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

Step 7.1.5.14

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

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

Step 7.1.5.15

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

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

Step 7.1.5.16

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

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

Step 7.1.6

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

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

Step 7.2

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

$x + 2 = 0$

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

Step 7.3

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

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Step 7.3.1

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

$x + 2 = 0$

Step 7.3.2

Subtract $2$ from both sides of the equation.

$x = - 2$

Step 7.4

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

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Step 7.4.1

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

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

Step 7.4.2

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

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Step 7.4.2.1

Use the quadratic formula to find the solutions.

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

Step 7.4.2.2

Substitute the values $a = 1$ , $b = 2$ , and $c = - 1$ into the quadratic formula and solve for $x$ .

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

Step 7.4.2.3

Simplify.

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Step 7.4.2.3.1

Simplify the numerator.

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Step 7.4.2.3.1.1

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

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

Step 7.4.2.3.1.2

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

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Step 7.4.2.3.1.2.1

Multiply $- 4$ by $1$ .

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

Step 7.4.2.3.1.2.2

Multiply $- 4$ by $- 1$ .

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

Step 7.4.2.3.1.3

Add $4$ and $4$ .

$x = \frac{- 2 \pm \sqrt{8}}{2 \cdot 1}$

Step 7.4.2.3.1.4

Rewrite $8$ as $2^{2} \cdot 2$ .

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Step 7.4.2.3.1.4.1

Factor $4$ out of $8$ .

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

Step 7.4.2.3.1.4.2

Rewrite $4$ as $2^{2}$ .

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

Step 7.4.2.3.1.5

Pull terms out from under the radical.

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

Step 7.4.2.3.2

Multiply $2$ by $1$ .

$x = \frac{- 2 \pm 2 \sqrt{2}}{2}$

Step 7.4.2.3.3

Simplify $\frac{- 2 \pm 2 \sqrt{2}}{2}$ .

$x = - 1 \pm \sqrt{2}$

Step 7.4.2.4

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 7.4.2.4.1

Simplify the numerator.

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Step 7.4.2.4.1.1

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

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

Step 7.4.2.4.1.2

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

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Step 7.4.2.4.1.2.1

Multiply $- 4$ by $1$ .

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

Step 7.4.2.4.1.2.2

Multiply $- 4$ by $- 1$ .

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

Step 7.4.2.4.1.3

Add $4$ and $4$ .

$x = \frac{- 2 \pm \sqrt{8}}{2 \cdot 1}$

Step 7.4.2.4.1.4

Rewrite $8$ as $2^{2} \cdot 2$ .

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Step 7.4.2.4.1.4.1

Factor $4$ out of $8$ .

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

Step 7.4.2.4.1.4.2

Rewrite $4$ as $2^{2}$ .

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

Step 7.4.2.4.1.5

Pull terms out from under the radical.

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

Step 7.4.2.4.2

Multiply $2$ by $1$ .

$x = \frac{- 2 \pm 2 \sqrt{2}}{2}$

Step 7.4.2.4.3

Simplify $\frac{- 2 \pm 2 \sqrt{2}}{2}$ .

$x = - 1 \pm \sqrt{2}$

Step 7.4.2.4.4

Change the $\pm$ to $+$ .

$x = - 1 + \sqrt{2}$

Step 7.4.2.5

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Step 7.4.2.5.1

Simplify the numerator.

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Step 7.4.2.5.1.1

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

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

Step 7.4.2.5.1.2

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

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Step 7.4.2.5.1.2.1

Multiply $- 4$ by $1$ .

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

Step 7.4.2.5.1.2.2

Multiply $- 4$ by $- 1$ .

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

Step 7.4.2.5.1.3

Add $4$ and $4$ .

$x = \frac{- 2 \pm \sqrt{8}}{2 \cdot 1}$

Step 7.4.2.5.1.4

Rewrite $8$ as $2^{2} \cdot 2$ .

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Step 7.4.2.5.1.4.1

Factor $4$ out of $8$ .

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

Step 7.4.2.5.1.4.2

Rewrite $4$ as $2^{2}$ .

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

Step 7.4.2.5.1.5

Pull terms out from under the radical.

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

Step 7.4.2.5.2

Multiply $2$ by $1$ .

$x = \frac{- 2 \pm 2 \sqrt{2}}{2}$

Step 7.4.2.5.3

Simplify $\frac{- 2 \pm 2 \sqrt{2}}{2}$ .

$x = - 1 \pm \sqrt{2}$

Step 7.4.2.5.4

Change the $\pm$ to $-$ .

$x = - 1 - \sqrt{2}$

Step 7.4.2.6

The final answer is the combination of both solutions.

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

Step 7.5

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

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

Step 8

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

$(x - 2) (x + 2) (x + 1 - \sqrt{2}) (x + 1 + \sqrt{2})$

Step 9

These are the roots (zeros) of the polynomial $x^{4} + 2 x^{3} - 5 x^{2} - 8 x + 4$ .

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

Step 10

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$x = 2, - 2, 0.41421356 \dots, - 2.41421356 \dots$

Step 11