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

Find all rational zeros of the polynomial, and write the polynomial in factored form.
$P\left(x\right)=4{x}^{3}-7x+3$
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Step 1
Given:
$P\left(x\right)=4{x}^{3}-7x+3$
First check the solution of cubic equation by putting the different value of x.
If x=1
$P\left(1\right)=4{\left(1\right)}^{3}-7\left(1\right)+3=7-7=0$
So, x = 1 is the solution of given cubic equation.
$4{x}^{3}-7x+3$ is divided by x-1
$\frac{4{x}^{3}-7x+3}{x-1}=4{x}^{2}+4x-3$
Step 2
Factorize $4{x}^{2}+4x-3$
$4{x}^{2}-2x+6x-3=0$
2x(2x-1)+3(2x-1)=0
(2x-1)(2x+3)=0
(2x-1)=0 or (2x+3)=0
2x=1 or 2x=-3
$x=\frac{1}{2}$ or $x=-\frac{3}{2}$
So,
Zeroes of polynomial are: x=1, $x=\frac{1}{2},x=-\frac{3}{2}$