Precalculus Examples

$3 x^{4} - 26 x^{2} - 9$

Step 1

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

$3 {(x^{2})}^{2} - 26 x^{2} - 9$

Step 2

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

$3 u^{2} - 26 u - 9$

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 $a \cdot c = 3 \cdot - 9 = - 27$ and whose sum is $b = - 26$ .

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Step 3.1.1

Factor $- 26$ out of $- 26 u$ .

$3 u^{2} - 26 (u) - 9$

Step 3.1.2

Rewrite $- 26$ as $1$ plus $- 27$

$3 u^{2} + (1 - 27) u - 9$

Step 3.1.3

Apply the distributive property.

$3 u^{2} + 1 u - 27 u - 9$

Step 3.1.4

Multiply $u$ by $1$ .

$3 u^{2} + u - 27 u - 9$

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.

$(3 u^{2} + u) - 27 u - 9$

Step 3.2.2

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

$u (3 u + 1) - 9 (3 u + 1)$

Step 3.3

Factor the polynomial by factoring out the greatest common factor, $3 u + 1$ .

$(3 u + 1) (u - 9)$

Step 4

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

$(3 x^{2} + 1) (x^{2} - 9)$

Step 5

Rewrite $9$ as $3^{2}$ .

$(3 x^{2} + 1) (x^{2} - 3^{2})$

Step 6

Factor.

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Step 6.1

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 $b = 3$ .

$(3 x^{2} + 1) ((x + 3) (x - 3))$

Step 6.2

Remove unnecessary parentheses.

$(3 x^{2} + 1) (x + 3) (x - 3)$