Calculus Examples

$4 x^{3} - 36 x^{2} + 96 x - 80$

Step 1

Factor $4$ out of $4 x^{3} - 36 x^{2} + 96 x - 80$ .

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Step 1.1

Factor $4$ out of $4 x^{3}$ .

$4 (x^{3}) - 36 x^{2} + 96 x - 80$

Step 1.2

Factor $4$ out of $- 36 x^{2}$ .

$4 (x^{3}) + 4 (- 9 x^{2}) + 96 x - 80$

Step 1.3

Factor $4$ out of $96 x$ .

$4 (x^{3}) + 4 (- 9 x^{2}) + 4 (24 x) - 80$

Step 1.4

Factor $4$ out of $- 80$ .

$4 x^{3} + 4 (- 9 x^{2}) + 4 (24 x) + 4 \cdot - 20$

Step 1.5

Factor $4$ out of $4 x^{3} + 4 (- 9 x^{2})$ .

$4 (x^{3} - 9 x^{2}) + 4 (24 x) + 4 \cdot - 20$

Step 1.6

Factor $4$ out of $4 (x^{3} - 9 x^{2}) + 4 (24 x)$ .

$4 (x^{3} - 9 x^{2} + 24 x) + 4 \cdot - 20$

Step 1.7

Factor $4$ out of $4 (x^{3} - 9 x^{2} + 24 x) + 4 \cdot - 20$ .

$4 (x^{3} - 9 x^{2} + 24 x - 20)$

Step 2

Factor $x^{3} - 9 x^{2} + 24 x - 20$ using the rational roots test.

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Step 2.1

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.

$p = \pm 1, \pm 20, \pm 2, \pm 10, \pm 4, \pm 5$

$q = \pm 1$

Step 2.2

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

$\pm 1, \pm 20, \pm 2, \pm 10, \pm 4, \pm 5$

Step 2.3

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 2.3.1

Substitute $2$ into the polynomial.

$2^{3} - 9 \cdot 2^{2} + 24 \cdot 2 - 20$

Step 2.3.2

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

$8 - 9 \cdot 2^{2} + 24 \cdot 2 - 20$

Step 2.3.3

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

$8 - 9 \cdot 4 + 24 \cdot 2 - 20$

Step 2.3.4

Multiply $- 9$ by $4$ .

$8 - 36 + 24 \cdot 2 - 20$

Step 2.3.5

Subtract $36$ from $8$ .

$- 28 + 24 \cdot 2 - 20$

Step 2.3.6

Multiply $24$ by $2$ .

$- 28 + 48 - 20$

Step 2.3.7

Add $- 28$ and $48$ .

$20 - 20$

Step 2.3.8

Subtract $20$ from $20$ .

$0$

Step 2.4

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.

$\frac{x^{3} - 9 x^{2} + 24 x - 20}{x - 2}$

Step 2.5

Divide $x^{3} - 9 x^{2} + 24 x - 20$ by $x - 2$ .

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Step 2.5.1

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

$x$

$2$

$x^{3}$

$9 x^{2}$

$24 x$

$20$

Step 2.5.2

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

					$x^{2}$
	$x$	-	$2$		$x^{3}$	-	$9 x^{2}$	+	$24 x$	-	$20$

Step 2.5.3

Multiply the new quotient term by the divisor.

				$x^{2}$
$x$	-	$2$		$x^{3}$	-	$9 x^{2}$	+	$24 x$	-	$20$
			+	$x^{3}$	-	$2 x^{2}$

Step 2.5.4

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

				$x^{2}$
$x$	-	$2$		$x^{3}$	-	$9 x^{2}$	+	$24 x$	-	$20$
			-	$x^{3}$	+	$2 x^{2}$

Step 2.5.5

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

				$x^{2}$
$x$	-	$2$		$x^{3}$	-	$9 x^{2}$	+	$24 x$	-	$20$
			-	$x^{3}$	+	$2 x^{2}$
					-	$7 x^{2}$

Step 2.5.6

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

				$x^{2}$
$x$	-	$2$		$x^{3}$	-	$9 x^{2}$	+	$24 x$	-	$20$
			-	$x^{3}$	+	$2 x^{2}$
					-	$7 x^{2}$	+	$24 x$

Step 2.5.7

Divide the highest order term in the dividend $- 7 x^{2}$ by the highest order term in divisor $x$ .

				$x^{2}$	-	$7 x$
$x$	-	$2$		$x^{3}$	-	$9 x^{2}$	+	$24 x$	-	$20$
			-	$x^{3}$	+	$2 x^{2}$
					-	$7 x^{2}$	+	$24 x$

Step 2.5.8

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$7 x$
$x$	-	$2$		$x^{3}$	-	$9 x^{2}$	+	$24 x$	-	$20$
			-	$x^{3}$	+	$2 x^{2}$
					-	$7 x^{2}$	+	$24 x$
					-	$7 x^{2}$	+	$14 x$

Step 2.5.9

The expression needs to be subtracted from the dividend, so change all the signs in $- 7 x^{2} + 14 x$

				$x^{2}$	-	$7 x$
$x$	-	$2$		$x^{3}$	-	$9 x^{2}$	+	$24 x$	-	$20$
			-	$x^{3}$	+	$2 x^{2}$
					-	$7 x^{2}$	+	$24 x$
					+	$7 x^{2}$	-	$14 x$

Step 2.5.10

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

				$x^{2}$	-	$7 x$
$x$	-	$2$		$x^{3}$	-	$9 x^{2}$	+	$24 x$	-	$20$
			-	$x^{3}$	+	$2 x^{2}$
					-	$7 x^{2}$	+	$24 x$
					+	$7 x^{2}$	-	$14 x$
							+	$10 x$

Step 2.5.11

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

				$x^{2}$	-	$7 x$
$x$	-	$2$		$x^{3}$	-	$9 x^{2}$	+	$24 x$	-	$20$
			-	$x^{3}$	+	$2 x^{2}$
					-	$7 x^{2}$	+	$24 x$
					+	$7 x^{2}$	-	$14 x$
							+	$10 x$	-	$20$

Step 2.5.12

Divide the highest order term in the dividend $10 x$ by the highest order term in divisor $x$ .

				$x^{2}$	-	$7 x$	+	$10$
$x$	-	$2$		$x^{3}$	-	$9 x^{2}$	+	$24 x$	-	$20$
			-	$x^{3}$	+	$2 x^{2}$
					-	$7 x^{2}$	+	$24 x$
					+	$7 x^{2}$	-	$14 x$
							+	$10 x$	-	$20$

Step 2.5.13

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$7 x$	+	$10$
$x$	-	$2$		$x^{3}$	-	$9 x^{2}$	+	$24 x$	-	$20$
			-	$x^{3}$	+	$2 x^{2}$
					-	$7 x^{2}$	+	$24 x$
					+	$7 x^{2}$	-	$14 x$
							+	$10 x$	-	$20$
							+	$10 x$	-	$20$

Step 2.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $10 x - 20$

				$x^{2}$	-	$7 x$	+	$10$
$x$	-	$2$		$x^{3}$	-	$9 x^{2}$	+	$24 x$	-	$20$
			-	$x^{3}$	+	$2 x^{2}$
					-	$7 x^{2}$	+	$24 x$
					+	$7 x^{2}$	-	$14 x$
							+	$10 x$	-	$20$
							-	$10 x$	+	$20$

Step 2.5.15

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

				$x^{2}$	-	$7 x$	+	$10$
$x$	-	$2$		$x^{3}$	-	$9 x^{2}$	+	$24 x$	-	$20$
			-	$x^{3}$	+	$2 x^{2}$
					-	$7 x^{2}$	+	$24 x$
					+	$7 x^{2}$	-	$14 x$
							+	$10 x$	-	$20$
							-	$10 x$	+	$20$
										$0$

Step 2.5.16

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

$x^{2} - 7 x + 10$

Step 2.6

Write $x^{3} - 9 x^{2} + 24 x - 20$ as a set of factors.

$4 ((x - 2) (x^{2} - 7 x + 10))$

Step 3

Factor.

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Step 3.1

Factor $x^{2} - 7 x + 10$ using the AC method.

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Step 3.1.1

Factor $x^{2} - 7 x + 10$ using the AC method.

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Step 3.1.1.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $10$ and whose sum is $- 7$ .

$- 5, - 2$

Step 3.1.1.2

Write the factored form using these integers.

$4 ((x - 2) ((x - 5) (x - 2)))$

Step 3.1.2

Remove unnecessary parentheses.

$4 ((x - 2) (x - 5) (x - 2))$

Step 3.2

Remove unnecessary parentheses.

$4 (x - 2) (x - 5) (x - 2)$

Step 4

Combine exponents.

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Step 4.1

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

$4 ({(x - 2)}^{1} (x - 2)) (x - 5)$

Step 4.2

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

$4 ({(x - 2)}^{1} {(x - 2)}^{1}) (x - 5)$

Step 4.3

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

$4 {(x - 2)}^{1 + 1} (x - 5)$

Step 4.4

Add $1$ and $1$ .

$4 {(x - 2)}^{2} (x - 5)$