Precalculus Examples

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

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.

$p = \pm 1, \pm 2$

$q = \pm 1, \pm 2$

Step 2

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

$\pm 1, \pm \frac{1}{2}, \pm 2$

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.

$2 {(- 1)}^{3} + 5 {(- 1)}^{2} - 1 - 2$

Step 4

Simplify the expression. In this case, the expression is equal to $0$ so $x = - 1$ is a root of the polynomial.

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Step 4.1

Remove parentheses.

$2 {(- 1)}^{3} + 5 {(- 1)}^{2} - 1 - 2$

Step 4.2

Simplify each term.

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Step 4.2.1

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

$2 \cdot - 1 + 5 {(- 1)}^{2} - 1 - 2$

Step 4.2.2

Multiply $2$ by $- 1$ .

$- 2 + 5 {(- 1)}^{2} - 1 - 2$

Step 4.2.3

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

$- 2 + 5 \cdot 1 - 1 - 2$

Step 4.2.4

Multiply $5$ by $1$ .

$- 2 + 5 - 1 - 2$

Step 4.3

Simplify by adding and subtracting.

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Step 4.3.1

Add $- 2$ and $5$ .

$3 - 1 - 2$

Step 4.3.2

Subtract $1$ from $3$ .

$2 - 2$

Step 4.3.3

Subtract $2$ from $2$ .

$0$

Step 5

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.

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

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.

$- 1$	$2$	$5$	$1$	$- 2$

Step 6.2

The first number in the dividend $(2)$ is put into the first position of the result area (below the horizontal line).

$- 1$	$2$	$5$	$1$	$- 2$

	$2$

Step 6.3

Multiply the newest entry in the result $(2)$ by the divisor $(- 1)$ and place the result of $(- 2)$ under the next term in the dividend $(5)$ .

$- 1$	$2$	$5$	$1$	$- 2$
		$- 2$
	$2$

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.

$- 1$	$2$	$5$	$1$	$- 2$
		$- 2$
	$2$	$3$

Step 6.5

Multiply the newest entry in the result $(3)$ by the divisor $(- 1)$ and place the result of $(- 3)$ under the next term in the dividend $(1)$ .

$- 1$	$2$	$5$	$1$	$- 2$
		$- 2$	$- 3$
	$2$	$3$

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.

$- 1$	$2$	$5$	$1$	$- 2$
		$- 2$	$- 3$
	$2$	$3$	$- 2$

Step 6.7

Multiply the newest entry in the result $(- 2)$ by the divisor $(- 1)$ and place the result of $(2)$ under the next term in the dividend $(- 2)$ .

$- 1$	$2$	$5$	$1$	$- 2$
		$- 2$	$- 3$	$2$
	$2$	$3$	$- 2$

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.

$- 1$	$2$	$5$	$1$	$- 2$
		$- 2$	$- 3$	$2$
	$2$	$3$	$- 2$	$0$

Step 6.9

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

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

Step 7

Factor by grouping.

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Step 7.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 2 \cdot - 2 = - 4$ and whose sum is $b = 3$ .

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Step 7.1.1

Factor $3$ out of $3 x$ .

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

Step 7.1.2

Rewrite $3$ as $- 1$ plus $4$

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

Step 7.1.3

Apply the distributive property.

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

Step 7.2

Factor out the greatest common factor from each group.

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Step 7.2.1

Group the first two terms and the last two terms.

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

Step 7.2.2

Factor out the greatest common factor (GCF) from each group.

$(x + 1) + x (2 x - 1) + 2 (2 x - 1)$

Step 7.3

Factor the polynomial by factoring out the greatest common factor, $2 x - 1$ .

$(x + 1) + (2 x - 1) (x + 2)$

$(x + 1) (2 x - 1) (x + 2)$

Step 8

Factor the left side of the equation.

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Step 8.1

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

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Step 8.1.1

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.

$p = \pm 1, \pm 2$

$q = \pm 1, \pm 2$

Step 8.1.2

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

$\pm 1, \pm 0.5, \pm 2$

Step 8.1.3

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 8.1.3.1

Substitute $- 1$ into the polynomial.

$2 {(- 1)}^{3} + 5 {(- 1)}^{2} - 1 - 2$

Step 8.1.3.2

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

$2 \cdot - 1 + 5 {(- 1)}^{2} - 1 - 2$

Step 8.1.3.3

Multiply $2$ by $- 1$ .

$- 2 + 5 {(- 1)}^{2} - 1 - 2$

Step 8.1.3.4

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

$- 2 + 5 \cdot 1 - 1 - 2$

Step 8.1.3.5

Multiply $5$ by $1$ .

$- 2 + 5 - 1 - 2$

Step 8.1.3.6

Add $- 2$ and $5$ .

$3 - 1 - 2$

Step 8.1.3.7

Subtract $1$ from $3$ .

$2 - 2$

Step 8.1.3.8

Subtract $2$ from $2$ .

$0$

Step 8.1.4

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.

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

Step 8.1.5

Divide $2 x^{3} + 5 x^{2} + x - 2$ by $x + 1$ .

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Step 8.1.5.1

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

$x$

$1$

$2 x^{3}$

$5 x^{2}$

$x$

$2$

Step 8.1.5.2

Divide the highest order term in the dividend $2 x^{3}$ by the highest order term in divisor $x$ .

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

Step 8.1.5.3

Multiply the new quotient term by the divisor.

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

Step 8.1.5.4

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

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

Step 8.1.5.5

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

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

Step 8.1.5.6

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

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

Step 8.1.5.7

Divide the highest order term in the dividend $3 x^{2}$ by the highest order term in divisor $x$ .

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

Step 8.1.5.8

Multiply the new quotient term by the divisor.

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

Step 8.1.5.9

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

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

Step 8.1.5.10

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

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

Step 8.1.5.11

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

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

Step 8.1.5.12

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

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

Step 8.1.5.13

Multiply the new quotient term by the divisor.

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

Step 8.1.5.14

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

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

Step 8.1.5.15

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

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

Step 8.1.5.16

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

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

Step 8.1.6

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

$(x + 1) (2 x^{2} + 3 x - 2) = 0$

Step 8.2

Factor by grouping.

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Step 8.2.1

Factor by grouping.

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Step 8.2.1.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 2 \cdot - 2 = - 4$ and whose sum is $b = 3$ .

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Step 8.2.1.1.1

Factor $3$ out of $3 x$ .

$(x + 1) (2 x^{2} + 3 (x) - 2) = 0$

Step 8.2.1.1.2

Rewrite $3$ as $- 1$ plus $4$

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

Step 8.2.1.1.3

Apply the distributive property.

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

Step 8.2.1.2

Factor out the greatest common factor from each group.

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Step 8.2.1.2.1

Group the first two terms and the last two terms.

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

Step 8.2.1.2.2

Factor out the greatest common factor (GCF) from each group.

$(x + 1) (x (2 x - 1) + 2 (2 x - 1)) = 0$

Step 8.2.1.3

Factor the polynomial by factoring out the greatest common factor, $2 x - 1$ .

$(x + 1) ((2 x - 1) (x + 2)) = 0$

Step 8.2.2

Remove unnecessary parentheses.

$(x + 1) (2 x - 1) (x + 2) = 0$

Step 9

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

$x + 1 = 0$

$2 x - 1 = 0$

$x + 2 = 0$

Step 10

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

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Step 10.1

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

$x + 1 = 0$

Step 10.2

Subtract $1$ from both sides of the equation.

$x = - 1$

Step 11

Set $2 x - 1$ equal to $0$ and solve for $x$ .

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Step 11.1

Set $2 x - 1$ equal to $0$ .

$2 x - 1 = 0$

Step 11.2

Solve $2 x - 1 = 0$ for $x$ .

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Step 11.2.1

Add $1$ to both sides of the equation.

$2 x = 1$

Step 11.2.2

Divide each term in $2 x = 1$ by $2$ and simplify.

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Step 11.2.2.1

Divide each term in $2 x = 1$ by $2$ .

$\frac{2 x}{2} = \frac{1}{2}$

Step 11.2.2.2

Simplify the left side.

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Step 11.2.2.2.1

Cancel the common factor of $2$ .

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Step 11.2.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{1}{2}$

Step 11.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{1}{2}$

Step 12

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

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Step 12.1

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

$x + 2 = 0$

Step 12.2

Subtract $2$ from both sides of the equation.

$x = - 2$

Step 13

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

$x = - 1, \frac{1}{2}, - 2$

Step 14