Algebra Examples

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

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.

$p = \pm 1$

$q = \pm 1, \pm 2$

Step 2

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

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

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.

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

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

Simplify each term.

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Step 4.1.1

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

$2 \cdot - 1 + 3 {(- 1)}^{2} - 1$

Step 4.1.2

Multiply $2$ by $- 1$ .

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

Step 4.1.3

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

$- 2 + 3 \cdot 1 - 1$

Step 4.1.4

Multiply $3$ by $1$ .

$- 2 + 3 - 1$

Step 4.2

Simplify by adding and subtracting.

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Step 4.2.1

Add $- 2$ and $3$ .

$1 - 1$

Step 4.2.2

Subtract $1$ from $1$ .

$0$

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.

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

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.

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

Step 6.2

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

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

	$2$

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 $(3)$ .

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

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.

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

Step 6.5

Multiply the newest entry in the result $(1)$ by the divisor $(- 1)$ and place the result of $(- 1)$ under the next term in the dividend $(0)$ .

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

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.

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

Step 6.7

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

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

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.

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

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.

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

Step 6.10

Simplify the quotient polynomial.

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

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 $a \cdot c = 2 \cdot - 1 = - 2$ and whose sum is $b = 1$ .

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Step 7.1.1

Multiply by $1$ .

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

Step 7.1.2

Rewrite $1$ as $- 1$ plus $2$

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

Step 7.1.3

Apply the distributive property.

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

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.

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

Step 7.2.2

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

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

Step 7.3

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

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

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

Step 8

Factor the left side of the equation.

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Step 8.1

Factor $2 x^{3} + 3 x^{2} - 1$ 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.

$p = \pm 1$

$q = \pm 1, \pm 2$

Step 8.1.2

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

$\pm 1, \pm 0.5$

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.

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

Step 8.1.3.2

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

$2 \cdot - 1 + 3 {(- 1)}^{2} - 1$

Step 8.1.3.3

Multiply $2$ by $- 1$ .

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

Step 8.1.3.4

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

$- 2 + 3 \cdot 1 - 1$

Step 8.1.3.5

Multiply $3$ by $1$ .

$- 2 + 3 - 1$

Step 8.1.3.6

Add $- 2$ and $3$ .

$1 - 1$

Step 8.1.3.7

Subtract $1$ from $1$ .

$0$

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.

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

Step 8.1.5

Divide $2 x^{3} + 3 x^{2} - 1$ 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 $0$ .

$x$

$1$

$2 x^{3}$

$3 x^{2}$

$0 x$

$1$

Step 8.1.5.2

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

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

Step 8.1.5.3

Multiply the new quotient term by the divisor.

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

Step 8.1.5.4

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

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

Step 8.1.5.5

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

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

Step 8.1.5.6

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

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

Step 8.1.5.7

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

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

Step 8.1.5.8

Multiply the new quotient term by the divisor.

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

Step 8.1.5.9

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

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

Step 8.1.5.10

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

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

Step 8.1.5.11

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

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

Step 8.1.5.12

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

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

Step 8.1.5.13

Multiply the new quotient term by the divisor.

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

Step 8.1.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $- x - 1$

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

Step 8.1.5.15

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

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

Step 8.1.5.16

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

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

Step 8.1.6

Write $2 x^{3} + 3 x^{2} - 1$ as a set of factors.

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

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 $a \cdot c = 2 \cdot - 1 = - 2$ and whose sum is $b = 1$ .

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Step 8.2.1.1.1

Multiply by $1$ .

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

Step 8.2.1.1.2

Rewrite $1$ as $- 1$ plus $2$

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

Step 8.2.1.1.3

Apply the distributive property.

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

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.

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

Step 8.2.1.2.2

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

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

Step 8.2.1.3

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

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

Step 8.2.2

Remove unnecessary parentheses.

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

Step 9

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

$x + 1 = 0$

$2 x - 1 = 0$

$x + 1 = 0$

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

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

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

Step 13