Question

factor the polynomial function $f(x)=10x^3 - 17x^2 - 42x + 9$.
note: write $\left(x - \frac{a}{b}\
ight)$ as $(bx - a)$
the zeros are
$f(x) = $
question help: video written example

Explanation:

Step1: Apply Rational Root Theorem

Possible rational roots: $\pm1, \pm3, \pm9, \pm\frac{1}{2}, \pm\frac{3}{2}, \pm\frac{9}{2}, \pm\frac{1}{5}, \pm\frac{3}{5}, \pm\frac{9}{5}, \pm\frac{1}{10}, \pm\frac{3}{10}, \pm\frac{9}{10}$
Test $x=3$: $10(3)^3 -17(3)^2 -42(3)+9=270-153-126+9=0$
So $x=3$ is a zero, factor is $(x-3)$ or $(1x-3)$.

Step2: Polynomial Long Division

Divide $10x^3 -17x^2 -42x +9$ by $(x-3)$:

$$\begin{align*} 10x^3 -17x^2 -42x +9&=(x-3)(10x^2 +13x -3) \end{align*}$$

Step3: Factor quadratic

Factor $10x^2 +13x -3$:
Find two numbers: $15$ and $-2$, since $15\times(-2)=-30$, $15+(-2)=13$

$$\begin{align*} 10x^2 +13x -3&=10x^2 +15x -2x -3\\ &=5x(2x+3)-1(2x+3)\\ &=(5x-1)(2x+3) \end{align*}$$

Step4: Find all zeros

Set each factor to 0:
$x-3=0 \implies x=3$
$5x-1=0 \implies x=\frac{1}{5}$
$2x+3=0 \implies x=-\frac{3}{2}$

Answer:

The zeros are $\boldsymbol{3, \frac{1}{5}, -\frac{3}{2}}$
$f(x)=\boldsymbol{(x-3)(5x-1)(2x+3)}$