Calculus Examples

$f (x) = x^{3} - 5 x^{2} + 12 x - 8$

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.

$p = \pm 1, \pm 8, \pm 2, \pm 4$

$q = \pm 1$

Step 2

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

$\pm 1, \pm 8, \pm 2, \pm 4$

Step 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 3.1

Substitute $1$ into the polynomial.

$1^{3} - 5 \cdot 1^{2} + 12 \cdot 1 - 8$

Step 3.2

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

$1 - 5 \cdot 1^{2} + 12 \cdot 1 - 8$

Step 3.3

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

$1 - 5 \cdot 1 + 12 \cdot 1 - 8$

Step 3.4

Multiply $- 5$ by $1$ .

$1 - 5 + 12 \cdot 1 - 8$

Step 3.5

Subtract $5$ from $1$ .

$- 4 + 12 \cdot 1 - 8$

Step 3.6

Multiply $12$ by $1$ .

$- 4 + 12 - 8$

Step 3.7

Add $- 4$ and $12$ .

$8 - 8$

Step 3.8

Subtract $8$ from $8$ .

$0$

Step 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.

$\frac{x^{3} - 5 x^{2} + 12 x - 8}{x - 1}$

Step 5

Divide $x^{3} - 5 x^{2} + 12 x - 8$ by $x - 1$ .

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Step 5.1

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

$x$

$1$

$x^{3}$

$5 x^{2}$

$12 x$

$8$

Step 5.2

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

					$x^{2}$
	$x$	-	$1$		$x^{3}$	-	$5 x^{2}$	+	$12 x$	-	$8$

Step 5.3

Multiply the new quotient term by the divisor.

				$x^{2}$
$x$	-	$1$		$x^{3}$	-	$5 x^{2}$	+	$12 x$	-	$8$
			+	$x^{3}$	-	$x^{2}$

Step 5.4

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

				$x^{2}$
$x$	-	$1$		$x^{3}$	-	$5 x^{2}$	+	$12 x$	-	$8$
			-	$x^{3}$	+	$x^{2}$

Step 5.5

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

				$x^{2}$
$x$	-	$1$		$x^{3}$	-	$5 x^{2}$	+	$12 x$	-	$8$
			-	$x^{3}$	+	$x^{2}$
					-	$4 x^{2}$

Step 5.6

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

				$x^{2}$
$x$	-	$1$		$x^{3}$	-	$5 x^{2}$	+	$12 x$	-	$8$
			-	$x^{3}$	+	$x^{2}$
					-	$4 x^{2}$	+	$12 x$

Step 5.7

Divide the highest order term in the dividend $- 4 x^{2}$ by the highest order term in divisor $x$ .

				$x^{2}$	-	$4 x$
$x$	-	$1$		$x^{3}$	-	$5 x^{2}$	+	$12 x$	-	$8$
			-	$x^{3}$	+	$x^{2}$
					-	$4 x^{2}$	+	$12 x$

Step 5.8

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$4 x$
$x$	-	$1$		$x^{3}$	-	$5 x^{2}$	+	$12 x$	-	$8$
			-	$x^{3}$	+	$x^{2}$
					-	$4 x^{2}$	+	$12 x$
					-	$4 x^{2}$	+	$4 x$

Step 5.9

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

				$x^{2}$	-	$4 x$
$x$	-	$1$		$x^{3}$	-	$5 x^{2}$	+	$12 x$	-	$8$
			-	$x^{3}$	+	$x^{2}$
					-	$4 x^{2}$	+	$12 x$
					+	$4 x^{2}$	-	$4 x$

Step 5.10

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

				$x^{2}$	-	$4 x$
$x$	-	$1$		$x^{3}$	-	$5 x^{2}$	+	$12 x$	-	$8$
			-	$x^{3}$	+	$x^{2}$
					-	$4 x^{2}$	+	$12 x$
					+	$4 x^{2}$	-	$4 x$
							+	$8 x$

Step 5.11

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

				$x^{2}$	-	$4 x$
$x$	-	$1$		$x^{3}$	-	$5 x^{2}$	+	$12 x$	-	$8$
			-	$x^{3}$	+	$x^{2}$
					-	$4 x^{2}$	+	$12 x$
					+	$4 x^{2}$	-	$4 x$
							+	$8 x$	-	$8$

Step 5.12

Divide the highest order term in the dividend $8 x$ by the highest order term in divisor $x$ .

				$x^{2}$	-	$4 x$	+	$8$
$x$	-	$1$		$x^{3}$	-	$5 x^{2}$	+	$12 x$	-	$8$
			-	$x^{3}$	+	$x^{2}$
					-	$4 x^{2}$	+	$12 x$
					+	$4 x^{2}$	-	$4 x$
							+	$8 x$	-	$8$

Step 5.13

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$4 x$	+	$8$
$x$	-	$1$		$x^{3}$	-	$5 x^{2}$	+	$12 x$	-	$8$
			-	$x^{3}$	+	$x^{2}$
					-	$4 x^{2}$	+	$12 x$
					+	$4 x^{2}$	-	$4 x$
							+	$8 x$	-	$8$
							+	$8 x$	-	$8$

Step 5.14

The expression needs to be subtracted from the dividend, so change all the signs in $8 x - 8$

				$x^{2}$	-	$4 x$	+	$8$
$x$	-	$1$		$x^{3}$	-	$5 x^{2}$	+	$12 x$	-	$8$
			-	$x^{3}$	+	$x^{2}$
					-	$4 x^{2}$	+	$12 x$
					+	$4 x^{2}$	-	$4 x$
							+	$8 x$	-	$8$
							-	$8 x$	+	$8$

Step 5.15

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

				$x^{2}$	-	$4 x$	+	$8$
$x$	-	$1$		$x^{3}$	-	$5 x^{2}$	+	$12 x$	-	$8$
			-	$x^{3}$	+	$x^{2}$
					-	$4 x^{2}$	+	$12 x$
					+	$4 x^{2}$	-	$4 x$
							+	$8 x$	-	$8$
							-	$8 x$	+	$8$
										$0$

Step 5.16

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

$x^{2} - 4 x + 8$

Step 6

Write $x^{3} - 5 x^{2} + 12 x - 8$ as a set of factors.

$f (x) = (x - 1) (x^{2} - 4 x + 8)$