Trigonometry Examples

$6 \sin^{2} (x) - 14 \sin (x) + 8$

Step 1

Factor $2$ out of $6 \sin^{2} (x) - 14 \sin (x) + 8$ .

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Step 1.1

Factor $2$ out of $6 \sin^{2} (x)$ .

$2 (3 \sin^{2} (x)) - 14 \sin (x) + 8$

Step 1.2

Factor $2$ out of $- 14 \sin (x)$ .

$2 (3 \sin^{2} (x)) + 2 (- 7 \sin (x)) + 8$

Step 1.3

Factor $2$ out of $8$ .

$2 (3 \sin^{2} (x)) + 2 (- 7 \sin (x)) + 2 (4)$

Step 1.4

Factor $2$ out of $2 (3 \sin^{2} (x)) + 2 (- 7 \sin (x))$ .

$2 (3 \sin^{2} (x) - 7 \sin (x)) + 2 (4)$

Step 1.5

Factor $2$ out of $2 (3 \sin^{2} (x) - 7 \sin (x)) + 2 (4)$ .

$2 (3 \sin^{2} (x) - 7 \sin (x) + 4)$

Step 2

Let $u = \sin (x)$ . Substitute $u$ for all occurrences of $\sin (x)$ .

$2 (3 u^{2} - 7 u + 4)$

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 $a \cdot c = 3 \cdot 4 = 12$ and whose sum is $b = - 7$ .

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Step 3.1.1

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

$2 (3 u^{2} - 7 (u) + 4)$

Step 3.1.2

Rewrite $- 7$ as $- 3$ plus $- 4$

$2 (3 u^{2} + (- 3 - 4) u + 4)$

Step 3.1.3

Apply the distributive property.

$2 (3 u^{2} - 3 u - 4 u + 4)$

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.

$2 ((3 u^{2} - 3 u) - 4 u + 4)$

Step 3.2.2

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

$2 (3 u (u - 1) - 4 (u - 1))$

Step 3.3

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

$2 ((u - 1) (3 u - 4))$

Step 4

Factor.

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Step 4.1

Replace all occurrences of $u$ with $\sin (x)$ .

$2 ((\sin (x) - 1) (3 \sin (x) - 4))$

Step 4.2

Remove unnecessary parentheses.

$2 (\sin (x) - 1) (3 \sin (x) - 4)$