Precalculus Examples

$x^{5} + 3 x^{4} - 25 x^{3} - 79 x^{2} + 100$

Step 1

Regroup terms.

$x^{5} - 25 x^{3} + 3 x^{4} - 79 x^{2} + 100$

Step 2

Factor $x^{3}$ out of $x^{5} - 25 x^{3}$ .

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Step 2.1

Factor $x^{3}$ out of $x^{5}$ .

$x^{3} x^{2} - 25 x^{3} + 3 x^{4} - 79 x^{2} + 100$

Step 2.2

Factor $x^{3}$ out of $- 25 x^{3}$ .

$x^{3} x^{2} + x^{3} \cdot - 25 + 3 x^{4} - 79 x^{2} + 100$

Step 2.3

Factor $x^{3}$ out of $x^{3} x^{2} + x^{3} \cdot - 25$ .

$x^{3} (x^{2} - 25) + 3 x^{4} - 79 x^{2} + 100$

Step 3

Rewrite $25$ as $5^{2}$ .

$x^{3} (x^{2} - 5^{2}) + 3 x^{4} - 79 x^{2} + 100$

Step 4

Factor.

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Step 4.1

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 5$ .

$x^{3} ((x + 5) (x - 5)) + 3 x^{4} - 79 x^{2} + 100$

Step 4.2

Remove unnecessary parentheses.

$x^{3} (x + 5) (x - 5) + 3 x^{4} - 79 x^{2} + 100$

Step 5

Rewrite $x^{4}$ as ${(x^{2})}^{2}$ .

$x^{3} (x + 5) (x - 5) + 3 {(x^{2})}^{2} - 79 x^{2} + 100$

Step 6

Let $u = x^{2}$ . Substitute $u$ for all occurrences of $x^{2}$ .

$x^{3} (x + 5) (x - 5) + 3 u^{2} - 79 u + 100$

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 $a \cdot c = 3 \cdot 100 = 300$ and whose sum is $b = - 79$ .

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Step 7.1.1

Factor $- 79$ out of $- 79 u$ .

$x^{3} (x + 5) (x - 5) + 3 u^{2} - 79 (u) + 100$

Step 7.1.2

Rewrite $- 79$ as $- 4$ plus $- 75$

$x^{3} (x + 5) (x - 5) + 3 u^{2} + (- 4 - 75) u + 100$

Step 7.1.3

Apply the distributive property.

$x^{3} (x + 5) (x - 5) + 3 u^{2} - 4 u - 75 u + 100$

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.

$x^{3} (x + 5) (x - 5) + (3 u^{2} - 4 u) - 75 u + 100$

Step 7.2.2

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

$x^{3} (x + 5) (x - 5) + u (3 u - 4) - 25 (3 u - 4)$

Step 7.3

Factor the polynomial by factoring out the greatest common factor, $3 u - 4$ .

$x^{3} (x + 5) (x - 5) + (3 u - 4) (u - 25)$

Step 8

Replace all occurrences of $u$ with $x^{2}$ .

$x^{3} (x + 5) (x - 5) + (3 x^{2} - 4) (x^{2} - 25)$

Step 9

Rewrite $25$ as $5^{2}$ .

$x^{3} (x + 5) (x - 5) + (3 x^{2} - 4) (x^{2} - 5^{2})$

Step 10

Factor.

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Step 10.1

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 5$ .

$x^{3} (x + 5) (x - 5) + (3 x^{2} - 4) ((x + 5) (x - 5))$

Step 10.2

Remove unnecessary parentheses.

$x^{3} (x + 5) (x - 5) + (3 x^{2} - 4) (x + 5) (x - 5)$

Step 11

Factor $(x + 5) (x - 5)$ out of $x^{3} (x + 5) (x - 5) + (3 x^{2} - 4) (x + 5) (x - 5)$ .

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Step 11.1

Factor $(x + 5) (x - 5)$ out of $x^{3} (x + 5) (x - 5)$ .

$(x + 5) (x - 5) (x^{3}) + (3 x^{2} - 4) (x + 5) (x - 5)$

Step 11.2

Factor $(x + 5) (x - 5)$ out of $(3 x^{2} - 4) (x + 5) (x - 5)$ .

$(x + 5) (x - 5) (x^{3}) + (x + 5) (x - 5) (3 x^{2} - 4)$

Step 11.3

Factor $(x + 5) (x - 5)$ out of $(x + 5) (x - 5) (x^{3}) + (x + 5) (x - 5) (3 x^{2} - 4)$ .

$(x + 5) (x - 5) (x^{3} + 3 x^{2} - 4)$

Step 12

Factor.

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Step 12.1

Rewrite $x^{3} + 3 x^{2} - 4$ in a factored form.

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Step 12.1.1

Factor $x^{3} + 3 x^{2} - 4$ using the rational roots test.

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Step 12.1.1.1

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.

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

$q = \pm 1$

Step 12.1.1.2

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

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

Step 12.1.1.3

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 12.1.1.3.1

Substitute $1$ into the polynomial.

$1^{3} + 3 \cdot 1^{2} - 4$

Step 12.1.1.3.2

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

$1 + 3 \cdot 1^{2} - 4$

Step 12.1.1.3.3

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

$1 + 3 \cdot 1 - 4$

Step 12.1.1.3.4

Multiply $3$ by $1$ .

$1 + 3 - 4$

Step 12.1.1.3.5

Add $1$ and $3$ .

$4 - 4$

Step 12.1.1.3.6

Subtract $4$ from $4$ .

$0$

Step 12.1.1.4

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.

$\frac{x^{3} + 3 x^{2} - 4}{x - 1}$

Step 12.1.1.5

Divide $x^{3} + 3 x^{2} - 4$ by $x - 1$ .

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Step 12.1.1.5.1

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

$x$

$1$

$x^{3}$

$3 x^{2}$

$0 x$

$4$

Step 12.1.1.5.2

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

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

Step 12.1.1.5.3

Multiply the new quotient term by the divisor.

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

Step 12.1.1.5.4

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

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

Step 12.1.1.5.5

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

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

Step 12.1.1.5.6

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

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

Step 12.1.1.5.7

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

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

Step 12.1.1.5.8

Multiply the new quotient term by the divisor.

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

Step 12.1.1.5.9

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

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

Step 12.1.1.5.10

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

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

Step 12.1.1.5.11

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

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

Step 12.1.1.5.12

Divide the highest order term in the dividend $4 x$ by the highest order term in divisor $x$ .

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

Step 12.1.1.5.13

Multiply the new quotient term by the divisor.

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

Step 12.1.1.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $4 x - 4$

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

Step 12.1.1.5.15

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

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

Step 12.1.1.5.16

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

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

Step 12.1.1.6

Write $x^{3} + 3 x^{2} - 4$ as a set of factors.

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

Step 12.1.2

Factor using the perfect square rule.

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Step 12.1.2.1

Rewrite $4$ as $2^{2}$ .

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

Step 12.1.2.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$4 x = 2 \cdot x \cdot 2$

Step 12.1.2.3

Rewrite the polynomial.

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

Step 12.1.2.4

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = x$ and $b = 2$ .

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

Step 12.2

Remove unnecessary parentheses.

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