# Find all rational zeros of the polynomial, and write the polynomial in factored form. P(x)=4x^{4}-37x^{2}+9

Find all rational zeros of the polynomial, and write the polynomial in factored form.
$P\left(x\right)=4{x}^{4}-37{x}^{2}+9$
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Step 1
Given, $P\left(x\right)=4{x}^{4}-37{x}^{2}+9$
For zeroes, substituting,P(x)=0
$⇒4{x}^{4}-37{x}^{2}+9=0$
$⇒4{x}^{4}-36{x}^{2}-{x}^{2}+9=0$
$⇒4{x}^{2}\left({x}^{2}-9\right)-\left({x}^{2}-9\right)=0$
$⇒\left({x}^{2}-9\right)\left(4{x}^{2}-1\right)=0$
$⇒\left({x}^{2}-{3}^{2}\right)\left({\left(2x\right)}^{2}-1\right)=0$
$⇒\left(x+3\right)\left(x-3\right)\left(2x+1\right)\left(2x-1\right)=0$
$\left(Using,\left({a}^{2}-{b}^{2}\right)=\left(a+b\right)\left(a-b\right)\right)$
$⇒\left(x+3\right)=0$, or, (x-3)=0, or , (2x+1)=0, or, (2x-1)=0
$⇒x=-3,3,-\frac{1}{2},\frac{1}{2}$
Hence, zeroes are: $x=\left(-3,-\frac{1}{2},\frac{1}{2},3\right)$.
Step 2
Required factorized form is:
$4{x}^{4}-37{x}^{2}+9=\left(x+3\right)\left(x-3\right)\left(2x+1\right)\left(2x-1\right)$