Precalculus Examples

$20 x^{4} - x^{2} - 1$

Step 1

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

$20 {(x^{2})}^{2} - x^{2} - 1$

Step 2

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

$20 u^{2} - u - 1$

Step 3

Factor by grouping.

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Step 3.1

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 $a \cdot c = 20 \cdot - 1 = - 20$ and whose sum is $b = - 1$ .

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Step 3.1.1

Factor $- 1$ out of $- u$ .

$20 u^{2} - (u) - 1$

Step 3.1.2

Rewrite $- 1$ as $4$ plus $- 5$

$20 u^{2} + (4 - 5) u - 1$

Step 3.1.3

Apply the distributive property.

$20 u^{2} + 4 u - 5 u - 1$

Step 3.2

Factor out the greatest common factor from each group.

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Step 3.2.1

Group the first two terms and the last two terms.

$(20 u^{2} + 4 u) - 5 u - 1$

Step 3.2.2

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

$4 u (5 u + 1) - (5 u + 1)$

Step 3.3

Factor the polynomial by factoring out the greatest common factor, $5 u + 1$ .

$(5 u + 1) (4 u - 1)$

Step 4

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

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

Step 5

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

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

Step 6

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

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

Step 7

Factor.

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Step 7.1

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 2 x$ and $b = 1$ .

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

Step 7.2

Remove unnecessary parentheses.

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