Calculus Examples

$2 - 15 x + 9 x^{2} - x^{3}$

Step 1

Reorder terms.

$- x^{3} + 9 x^{2} - 15 x + 2$

Step 2

Factor $- x^{3} + 9 x^{2} - 15 x + 2$ 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.

$p = \pm 1, \pm 2$

$q = \pm 1$

Step 2.2

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

$\pm 1, \pm 2$

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.

$- 2^{3} + 9 \cdot 2^{2} - 15 \cdot 2 + 2$

Step 2.3.2

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

$- 1 \cdot 8 + 9 \cdot 2^{2} - 15 \cdot 2 + 2$

Step 2.3.3

Multiply $- 1$ by $8$ .

$- 8 + 9 \cdot 2^{2} - 15 \cdot 2 + 2$

Step 2.3.4

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

$- 8 + 9 \cdot 4 - 15 \cdot 2 + 2$

Step 2.3.5

Multiply $9$ by $4$ .

$- 8 + 36 - 15 \cdot 2 + 2$

Step 2.3.6

Add $- 8$ and $36$ .

$28 - 15 \cdot 2 + 2$

Step 2.3.7

Multiply $- 15$ by $2$ .

$28 - 30 + 2$

Step 2.3.8

Subtract $30$ from $28$ .

$- 2 + 2$

Step 2.3.9

Add $- 2$ and $2$ .

$0$

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.

$\frac{- x^{3} + 9 x^{2} - 15 x + 2}{x - 2}$

Step 2.5

Divide $- x^{3} + 9 x^{2} - 15 x + 2$ 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 $0$ .

$x$

$2$

$x^{3}$

$9 x^{2}$

$15 x$

$2$

Step 2.5.2

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

				-	$x^{2}$
	$x$	-	$2$	-	$x^{3}$	+	$9 x^{2}$	-	$15 x$	+	$2$

Step 2.5.3

Multiply the new quotient term by the divisor.

			-	$x^{2}$
$x$	-	$2$	-	$x^{3}$	+	$9 x^{2}$	-	$15 x$	+	$2$
			-	$x^{3}$	+	$2 x^{2}$

Step 2.5.4

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

			-	$x^{2}$
$x$	-	$2$	-	$x^{3}$	+	$9 x^{2}$	-	$15 x$	+	$2$
			+	$x^{3}$	-	$2 x^{2}$

Step 2.5.5

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

			-	$x^{2}$
$x$	-	$2$	-	$x^{3}$	+	$9 x^{2}$	-	$15 x$	+	$2$
			+	$x^{3}$	-	$2 x^{2}$
					+	$7 x^{2}$

Step 2.5.6

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

			-	$x^{2}$
$x$	-	$2$	-	$x^{3}$	+	$9 x^{2}$	-	$15 x$	+	$2$
			+	$x^{3}$	-	$2 x^{2}$
					+	$7 x^{2}$	-	$15 x$

Step 2.5.7

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

			-	$x^{2}$	+	$7 x$
$x$	-	$2$	-	$x^{3}$	+	$9 x^{2}$	-	$15 x$	+	$2$
			+	$x^{3}$	-	$2 x^{2}$
					+	$7 x^{2}$	-	$15 x$

Step 2.5.8

Multiply the new quotient term by the divisor.

			-	$x^{2}$	+	$7 x$
$x$	-	$2$	-	$x^{3}$	+	$9 x^{2}$	-	$15 x$	+	$2$
			+	$x^{3}$	-	$2 x^{2}$
					+	$7 x^{2}$	-	$15 x$
					+	$7 x^{2}$	-	$14 x$

Step 2.5.9

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

			-	$x^{2}$	+	$7 x$
$x$	-	$2$	-	$x^{3}$	+	$9 x^{2}$	-	$15 x$	+	$2$
			+	$x^{3}$	-	$2 x^{2}$
					+	$7 x^{2}$	-	$15 x$
					-	$7 x^{2}$	+	$14 x$

Step 2.5.10

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

			-	$x^{2}$	+	$7 x$
$x$	-	$2$	-	$x^{3}$	+	$9 x^{2}$	-	$15 x$	+	$2$
			+	$x^{3}$	-	$2 x^{2}$
					+	$7 x^{2}$	-	$15 x$
					-	$7 x^{2}$	+	$14 x$
							-	$x$

Step 2.5.11

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

			-	$x^{2}$	+	$7 x$
$x$	-	$2$	-	$x^{3}$	+	$9 x^{2}$	-	$15 x$	+	$2$
			+	$x^{3}$	-	$2 x^{2}$
					+	$7 x^{2}$	-	$15 x$
					-	$7 x^{2}$	+	$14 x$
							-	$x$	+	$2$

Step 2.5.12

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

			-	$x^{2}$	+	$7 x$	-	$1$
$x$	-	$2$	-	$x^{3}$	+	$9 x^{2}$	-	$15 x$	+	$2$
			+	$x^{3}$	-	$2 x^{2}$
					+	$7 x^{2}$	-	$15 x$
					-	$7 x^{2}$	+	$14 x$
							-	$x$	+	$2$

Step 2.5.13

Multiply the new quotient term by the divisor.

			-	$x^{2}$	+	$7 x$	-	$1$
$x$	-	$2$	-	$x^{3}$	+	$9 x^{2}$	-	$15 x$	+	$2$
			+	$x^{3}$	-	$2 x^{2}$
					+	$7 x^{2}$	-	$15 x$
					-	$7 x^{2}$	+	$14 x$
							-	$x$	+	$2$
							-	$x$	+	$2$

Step 2.5.14

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

			-	$x^{2}$	+	$7 x$	-	$1$
$x$	-	$2$	-	$x^{3}$	+	$9 x^{2}$	-	$15 x$	+	$2$
			+	$x^{3}$	-	$2 x^{2}$
					+	$7 x^{2}$	-	$15 x$
					-	$7 x^{2}$	+	$14 x$
							-	$x$	+	$2$
							+	$x$	-	$2$

Step 2.5.15

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

			-	$x^{2}$	+	$7 x$	-	$1$
$x$	-	$2$	-	$x^{3}$	+	$9 x^{2}$	-	$15 x$	+	$2$
			+	$x^{3}$	-	$2 x^{2}$
					+	$7 x^{2}$	-	$15 x$
					-	$7 x^{2}$	+	$14 x$
							-	$x$	+	$2$
							+	$x$	-	$2$
										$0$

Step 2.5.16

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

$- x^{2} + 7 x - 1$

Step 2.6

Write $- x^{3} + 9 x^{2} - 15 x + 2$ as a set of factors.

$(x - 2) (- x^{2} + 7 x - 1)$

Step 3

Factor.

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Step 3.1

Factor $- 1$ out of $- x^{2} + 7 x - 1$ .

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Step 3.1.1

Factor $- 1$ out of $- x^{2}$ .

$(x - 2) (- (x^{2}) + 7 x - 1)$

Step 3.1.2

Factor $- 1$ out of $7 x$ .

$(x - 2) (- (x^{2}) - (- 7 x) - 1)$

Step 3.1.3

Rewrite $- 1$ as $- 1 (1)$ .

$(x - 2) (- (x^{2}) - (- 7 x) - 1 (1))$

Step 3.1.4

Factor $- 1$ out of $- (x^{2}) - (- 7 x)$ .

$(x - 2) (- (x^{2} - 7 x) - 1 (1))$

Step 3.1.5

Factor $- 1$ out of $- (x^{2} - 7 x) - 1 (1)$ .

$(x - 2) (- (x^{2} - 7 x + 1))$

Step 3.2

Remove unnecessary parentheses.

$(x - 2) \cdot - 1 (x^{2} - 7 x + 1)$

Step 4

Factor out negative.

$- ((x - 2) (x^{2} - 7 x + 1))$

Step 5

Remove unnecessary parentheses.

$- (x - 2) (x^{2} - 7 x + 1)$