Calculus Examples

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

Step 1

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

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Step 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 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 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 1.3.1

Substitute $- 1$ into the polynomial.

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

Step 1.3.2

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

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

Step 1.3.3

Multiply $2$ by $- 1$ .

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

Step 1.3.4

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

$- 2 - 3 \cdot 1 - 3 \cdot - 1 + 2$

Step 1.3.5

Multiply $- 3$ by $1$ .

$- 2 - 3 - 3 \cdot - 1 + 2$

Step 1.3.6

Subtract $3$ from $- 2$ .

$- 5 - 3 \cdot - 1 + 2$

Step 1.3.7

Multiply $- 3$ by $- 1$ .

$- 5 + 3 + 2$

Step 1.3.8

Add $- 5$ and $3$ .

$- 2 + 2$

Step 1.3.9

Add $- 2$ and $2$ .

$0$

Step 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.

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

Step 1.5

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

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Step 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 $0$ .

$x$

$1$

$2 x^{3}$

$3 x^{2}$

$3 x$

$2$

Step 1.5.2

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

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

Step 1.5.3

Multiply the new quotient term by the divisor.

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

Step 1.5.4

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

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

Step 1.5.5

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

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

Step 1.5.6

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

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

Step 1.5.7

Divide the highest order term in the dividend $- 5 x^{2}$ by the highest order term in divisor $x$ .

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

Step 1.5.8

Multiply the new quotient term by the divisor.

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

Step 1.5.9

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

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

Step 1.5.10

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

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

Step 1.5.11

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

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

Step 1.5.12

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

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

Step 1.5.13

Multiply the new quotient term by the divisor.

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

Step 1.5.14

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

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

Step 1.5.15

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

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

Step 1.5.16

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

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

Step 1.6

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

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

Step 2

Factor by grouping.

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Step 2.1

Factor by grouping.

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Step 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 $b = - 5$ .

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Step 2.1.1.1

Factor $- 5$ out of $- 5 x$ .

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

Step 2.1.1.2

Rewrite $- 5$ as $- 1$ plus $- 4$

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

Step 2.1.1.3

Apply the distributive property.

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

Step 2.1.2

Factor out the greatest common factor from each group.

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Step 2.1.2.1

Group the first two terms and the last two terms.

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

Step 2.1.2.2

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

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

Step 2.1.3

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

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

Step 2.2

Remove unnecessary parentheses.

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