Precalculus Examples

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

Step 1

Regroup terms.

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

Step 2

Factor $- 2 x$ out of $- 2 x^{3} - 8 x$ .

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Step 2.1

Factor $- 2 x$ out of $- 2 x^{3}$ .

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

Step 2.2

Factor $- 2 x$ out of $- 8 x$ .

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

Step 2.3

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

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

Step 3

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

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

Step 4

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

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

Step 5

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

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Step 5.1

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 $5$ .

$1, 4$

Step 5.2

Write the factored form using these integers.

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

Step 6

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

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

Step 7

Factor $x^{2} + 4$ out of $- 2 x (x^{2} + 4) + (x^{2} + 1) (x^{2} + 4)$ .

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Step 7.1

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

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

Step 7.2

Factor $x^{2} + 4$ out of $(x^{2} + 1) (x^{2} + 4)$ .

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

Step 7.3

Factor $x^{2} + 4$ out of $(x^{2} + 4) (- 2 x) + (x^{2} + 4) (x^{2} + 1)$ .

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

Step 8

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

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

Step 9

Factor using the perfect square rule.

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Step 9.1

Rearrange terms.

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

Step 9.2

Rewrite $1$ as $1^{2}$ .

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

Step 9.3

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

$2 u = 2 \cdot u \cdot 1$

Step 9.4

Rewrite the polynomial.

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

Step 9.5

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = u$ and $b = 1$ .

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

Step 10

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

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