Precalculus Examples

$3 \sin^{2} (x) - 5 \sin (x) - 2$

Step 1

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

$3 u^{2} - 5 u - 2$

Step 2

Factor by grouping.

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Step 2.1

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 $a \cdot c = 3 \cdot - 2 = - 6$ and whose sum is $b = - 5$ .

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Step 2.1.1

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

$3 u^{2} - 5 (u) - 2$

Step 2.1.2

Rewrite $- 5$ as $1$ plus $- 6$

$3 u^{2} + (1 - 6) u - 2$

Step 2.1.3

Apply the distributive property.

$3 u^{2} + 1 u - 6 u - 2$

Step 2.1.4

Multiply $u$ by $1$ .

$3 u^{2} + u - 6 u - 2$

Step 2.2

Factor out the greatest common factor from each group.

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Step 2.2.1

Group the first two terms and the last two terms.

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

Step 2.2.2

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

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

Step 2.3

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

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

Step 3

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

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