Precalculus Examples

$- 6 x^{8} + 7 x^{4} + 5$

Step 1

Factor $- 1$ out of $- 6 x^{8} + 7 x^{4} + 5$ .

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Step 1.1

Factor $- 1$ out of $- 6 x^{8}$ .

$- (6 x^{8}) + 7 x^{4} + 5$

Step 1.2

Factor $- 1$ out of $7 x^{4}$ .

$- (6 x^{8}) - (- 7 x^{4}) + 5$

Step 1.3

Rewrite $5$ as $- 1 (- 5)$ .

$- (6 x^{8}) - (- 7 x^{4}) - 1 (- 5)$

Step 1.4

Factor $- 1$ out of $- (6 x^{8}) - (- 7 x^{4})$ .

$- (6 x^{8} - 7 x^{4}) - 1 (- 5)$

Step 1.5

Factor $- 1$ out of $- (6 x^{8} - 7 x^{4}) - 1 (- 5)$ .

$- (6 x^{8} - 7 x^{4} - 5)$

Step 2

Rewrite $x^{8}$ as ${(x^{4})}^{2}$ .

$- (6 {(x^{4})}^{2} - 7 x^{4} - 5)$

Step 3

Let $u = x^{4}$ . Substitute $u$ for all occurrences of $x^{4}$ .

$- (6 u^{2} - 7 u - 5)$

Step 4

Factor by grouping.

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Step 4.1

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 $a \cdot c = 6 \cdot - 5 = - 30$ and whose sum is $b = - 7$ .

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Step 4.1.1

Factor $- 7$ out of $- 7 u$ .

$- (6 u^{2} - 7 (u) - 5)$

Step 4.1.2

Rewrite $- 7$ as $3$ plus $- 10$

$- (6 u^{2} + (3 - 10) u - 5)$

Step 4.1.3

Apply the distributive property.

$- (6 u^{2} + 3 u - 10 u - 5)$

Step 4.2

Factor out the greatest common factor from each group.

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Step 4.2.1

Group the first two terms and the last two terms.

$- ((6 u^{2} + 3 u) - 10 u - 5)$

Step 4.2.2

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

$- (3 u (2 u + 1) - 5 (2 u + 1))$

Step 4.3

Factor the polynomial by factoring out the greatest common factor, $2 u + 1$ .

$- ((2 u + 1) (3 u - 5))$

Step 5

Factor.

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Step 5.1

Replace all occurrences of $u$ with $x^{4}$ .

$- ((2 x^{4} + 1) (3 x^{4} - 5))$

Step 5.2

Remove unnecessary parentheses.

$- (2 x^{4} + 1) (3 x^{4} - 5)$