# Find all rational zeros of the polynomial, and write the polynomial in factored form. P(x)=6x^{4}-7x^{3}-12x^{2}+3x+2

Find all rational zeros of the polynomial, and write the polynomial in factored form.
$P\left(x\right)=6{x}^{4}-7{x}^{3}-12{x}^{2}+3x+2$
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davonliefI
Given
$P\left(x\right)=6{x}^{4}-7{x}^{3}-12{x}^{2}+3x+2$
$P\left(x\right)=6{x}^{4}-7{x}^{3}-12{x}^{2}+3x+2$
use long division to find zeros of p(x)
from the long division, we get
x=-1, x=2 and $6{x}^{2}-x-1=0$
$x=\frac{-\left(-1\right)±\sqrt{{\left(-1\right)}^{2}-4\left(6\right)\left(-1\right)}}{2\left(6\right)}$
$=\frac{1±\sqrt{1+24}}{12}$
$=\frac{1±\sqrt{25}}{12}$
$=\frac{1±5}{12}$

Therefore the zeros of p(x) is x =-1, $-\frac{1}{3},\frac{1}{2}$ and 2
Factor of p(x),
$p\left(x\right)=\left(x+1\right)\left(x+\frac{1}{3}\right)\left(x-\frac{1}{2}\right)\left(x-2\right)$