# How do you factor and simplify 3sin^2y-7siny-20?

Aug 9, 2018

color(purple)(=> (3 x + 5) (x - 4)

#### Explanation:

$3 {\sin}^{2} y - 7 \sin y - 20$

Let $\sin y = x$

$\implies 3 {x}^{2} - 7 x - 20$

$\implies 3 {x}^{2} - 12 x + 5 x - 20$

$\implies 3 x \left(x - 4\right) + 5 \left(x - 4\right)$

color(purple)(=> (3 x + 5) (x - 4)

Aug 9, 2018

$\left(\sin y - 4\right) \left(3 \sin y + 5\right)$

#### Explanation:

$\text{this is a quadratic in sine and is factored in the same}$
$\text{way as a normal quadratic}$

$\text{the factors of the product } 3 \times - 20 = - 60$

$\text{which sum to "-7" are "-12" and } + 5$

$\text{split the middle term using these factors}$

$3 {\sin}^{2} y - 12 \sin y + 5 \sin y - 20 \leftarrow \textcolor{b l u e}{\text{factor by grouping}}$

$= \textcolor{red}{3 \sin y} \left(\sin y - 4\right) \textcolor{red}{+ 5} \left(\sin y - 4\right)$

$\text{take out the "color(blue)"common factor } \left(\sin y - 4\right)$

$= \left(\sin y - 4\right) \left(\textcolor{red}{3 \sin y + 5}\right)$

$3 {\sin}^{2} y - 7 \sin y - 20 = \left(\sin y - 4\right) \left(3 \sin y + 5\right)$