Precalculus Examples

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

Step 1

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

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

Step 2

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

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

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 $a \cdot c = 4 \cdot 1 = 4$ and whose sum is $b = - 5$ .

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Step 3.1.1

Factor $- 5$ out of $- 5 u$ .

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

Step 3.1.2

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

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

Step 3.1.3

Apply the distributive property.

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

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.

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

Step 3.2.2

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

$u (4 u - 1) - (4 u - 1)$

Step 3.3

Factor the polynomial by factoring out the greatest common factor, $4 u - 1$ .

$(4 u - 1) (u - 1)$

Step 4

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

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

Step 5

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

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

Step 6

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

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

Step 7

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 $b = 1$ .

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

Step 8

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

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

Step 9

Factor.

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Step 9.1

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 $b = 1$ .

$(2 x + 1) (2 x - 1) ((x + 1) (x - 1))$

Step 9.2

Remove unnecessary parentheses.

$(2 x + 1) (2 x - 1) (x + 1) (x - 1)$