Precalculus Examples

$x^{4} - 5 x^{3} + 8 x^{2} - 20 x + 16$

Step 1

Regroup terms.

$- 5 x^{3} - 20 x + x^{4} + 8 x^{2} + 16$

Step 2

Factor $- 5 x$ out of $- 5 x^{3} - 20 x$ .

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Step 2.1

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

$- 5 x (x^{2}) - 20 x + x^{4} + 8 x^{2} + 16$

Step 2.2

Factor $- 5 x$ out of $- 20 x$ .

$- 5 x (x^{2}) - 5 x (4) + x^{4} + 8 x^{2} + 16$

Step 2.3

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

$- 5 x (x^{2} + 4) + x^{4} + 8 x^{2} + 16$

Step 3

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

$- 5 x (x^{2} + 4) + {(x^{2})}^{2} + 8 x^{2} + 16$

Step 4

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

$- 5 x (x^{2} + 4) + u^{2} + 8 u + 16$

Step 5

Factor using the perfect square rule.

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Step 5.1

Rewrite $16$ as $4^{2}$ .

$- 5 x (x^{2} + 4) + u^{2} + 8 u + 4^{2}$

Step 5.2

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

$8 u = 2 \cdot u \cdot 4$

Step 5.3

Rewrite the polynomial.

$- 5 x (x^{2} + 4) + u^{2} + 2 \cdot u \cdot 4 + 4^{2}$

Step 5.4

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = u$ and $b = 4$ .

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

Step 6

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

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

Step 7

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

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Step 7.1

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

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

Step 7.2

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

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

Step 7.3

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

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

Step 8

Let $u = x$ . Substitute $u$ for all occurrences of $x$ .

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

Step 9

Factor $- 5 u + u^{2} + 4$ using the AC method.

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Step 9.1

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 $4$ and whose sum is $- 5$ .

$- 4, - 1$

Step 9.2

Write the factored form using these integers.

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

Step 10

Factor.

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Step 10.1

Replace all occurrences of $u$ with $x$ .

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

Step 10.2

Remove unnecessary parentheses.

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