Trigonometry Examples

$f (x) = 60 x^{4} + 86 x^{3} - 46 x^{2} - 43 x + 8$

Step 1

Regroup terms.

$f (x) = 86 x^{3} - 43 x + 60 x^{4} - 46 x^{2} + 8$

Step 2

Factor $43 x$ out of $86 x^{3} - 43 x$ .

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Step 2.1

Factor $43 x$ out of $86 x^{3}$ .

$f (x) = 43 x (2 x^{2}) - 43 x + 60 x^{4} - 46 x^{2} + 8$

Step 2.2

Factor $43 x$ out of $- 43 x$ .

$f (x) = 43 x (2 x^{2}) + 43 x (- 1) + 60 x^{4} - 46 x^{2} + 8$

Step 2.3

Factor $43 x$ out of $43 x (2 x^{2}) + 43 x (- 1)$ .

$f (x) = 43 x (2 x^{2} - 1) + 60 x^{4} - 46 x^{2} + 8$

Step 3

Factor $2$ out of $60 x^{4} - 46 x^{2} + 8$ .

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Step 3.1

Factor $2$ out of $60 x^{4}$ .

$f (x) = 43 x (2 x^{2} - 1) + 2 (30 x^{4}) - 46 x^{2} + 8$

Step 3.2

Factor $2$ out of $- 46 x^{2}$ .

$f (x) = 43 x (2 x^{2} - 1) + 2 (30 x^{4}) + 2 (- 23 x^{2}) + 8$

Step 3.3

Factor $2$ out of $8$ .

$f (x) = 43 x (2 x^{2} - 1) + 2 (30 x^{4}) + 2 (- 23 x^{2}) + 2 (4)$

Step 3.4

Factor $2$ out of $2 (30 x^{4}) + 2 (- 23 x^{2})$ .

$f (x) = 43 x (2 x^{2} - 1) + 2 (30 x^{4} - 23 x^{2}) + 2 (4)$

Step 3.5

Factor $2$ out of $2 (30 x^{4} - 23 x^{2}) + 2 (4)$ .

$f (x) = 43 x (2 x^{2} - 1) + 2 (30 x^{4} - 23 x^{2} + 4)$

Step 4

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

$f (x) = 43 x (2 x^{2} - 1) + 2 (30 {(x^{2})}^{2} - 23 x^{2} + 4)$

Step 5

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

$f (x) = 43 x (2 x^{2} - 1) + 2 (30 u^{2} - 23 u + 4)$

Step 6

Factor by grouping.

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Step 6.1

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 $a \cdot c = 30 \cdot 4 = 120$ and whose sum is $b = - 23$ .

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Step 6.1.1

Factor $- 23$ out of $- 23 u$ .

$f (x) = 43 x (2 x^{2} - 1) + 2 (30 u^{2} - 23 u + 4)$

Step 6.1.2

Rewrite $- 23$ as $- 8$ plus $- 15$

$f (x) = 43 x (2 x^{2} - 1) + 2 (30 u^{2} + (- 8 - 15) u + 4)$

Step 6.1.3

Apply the distributive property.

$f (x) = 43 x (2 x^{2} - 1) + 2 (30 u^{2} - 8 u - 15 u + 4)$

Step 6.2

Factor out the greatest common factor from each group.

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Step 6.2.1

Group the first two terms and the last two terms.

$f (x) = 43 x (2 x^{2} - 1) + 2 ((30 u^{2} - 8 u) - 15 u + 4)$

Step 6.2.2

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

$f (x) = 43 x (2 x^{2} - 1) + 2 (2 u (15 u - 4) - (15 u - 4))$

Step 6.3

Factor the polynomial by factoring out the greatest common factor, $15 u - 4$ .

$f (x) = 43 x (2 x^{2} - 1) + 2 ((15 u - 4) (2 u - 1))$

Step 7

Factor.

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Step 7.1

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

$f (x) = 43 x (2 x^{2} - 1) + 2 ((15 x^{2} - 4) (2 x^{2} - 1))$

Step 7.2

Remove unnecessary parentheses.

$f (x) = 43 x (2 x^{2} - 1) + 2 (15 x^{2} - 4) (2 x^{2} - 1)$

Step 8

Factor $2 x^{2} - 1$ out of $43 x (2 x^{2} - 1) + 2 (15 x^{2} - 4) (2 x^{2} - 1)$ .

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Step 8.1

Factor $2 x^{2} - 1$ out of $43 x (2 x^{2} - 1)$ .

$f (x) = (2 x^{2} - 1) (43 x) + 2 (15 x^{2} - 4) (2 x^{2} - 1)$

Step 8.2

Factor $2 x^{2} - 1$ out of $2 (15 x^{2} - 4) (2 x^{2} - 1)$ .

$f (x) = (2 x^{2} - 1) (43 x) + (2 x^{2} - 1) (2 (15 x^{2} - 4))$

Step 8.3

Factor $2 x^{2} - 1$ out of $(2 x^{2} - 1) (43 x) + (2 x^{2} - 1) (2 (15 x^{2} - 4))$ .

$f (x) = (2 x^{2} - 1) (43 x + 2 (15 x^{2} - 4))$

Step 9

Apply the distributive property.

$f (x) = (2 x^{2} - 1) (43 x + 2 (15 x^{2}) + 2 \cdot - 4)$

Step 10

Multiply $15$ by $2$ .

$f (x) = (2 x^{2} - 1) (43 x + 30 x^{2} + 2 \cdot - 4)$

Step 11

Multiply $2$ by $- 4$ .

$f (x) = (2 x^{2} - 1) (43 x + 30 x^{2} - 8)$

Step 12

Reorder terms.

$f (x) = (2 x^{2} - 1) (30 x^{2} + 43 x - 8)$

Step 13

Factor.

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Step 13.1

Factor by grouping.

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Step 13.1.1

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 $a \cdot c = 30 \cdot - 8 = - 240$ and whose sum is $b = 43$ .

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Step 13.1.1.1

Factor $43$ out of $43 x$ .

$f (x) = (2 x^{2} - 1) (30 x^{2} + 43 (x) - 8)$

Step 13.1.1.2

Rewrite $43$ as $- 5$ plus $48$

$f (x) = (2 x^{2} - 1) (30 x^{2} + (- 5 + 48) x - 8)$

Step 13.1.1.3

Apply the distributive property.

$f (x) = (2 x^{2} - 1) (30 x^{2} - 5 x + 48 x - 8)$

Step 13.1.2

Factor out the greatest common factor from each group.

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Step 13.1.2.1

Group the first two terms and the last two terms.

$f (x) = (2 x^{2} - 1) ((30 x^{2} - 5 x) + 48 x - 8)$

Step 13.1.2.2

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

$f (x) = (2 x^{2} - 1) (5 x (6 x - 1) + 8 (6 x - 1))$

Step 13.1.3

Factor the polynomial by factoring out the greatest common factor, $6 x - 1$ .

$f (x) = (2 x^{2} - 1) ((6 x - 1) (5 x + 8))$

Step 13.2

Remove unnecessary parentheses.

$f (x) = (2 x^{2} - 1) (6 x - 1) (5 x + 8)$