Sam Longoria

2021-12-08

Find the complete factorization and all five zeros of the polynomial
$P\left(x\right)=3{x}^{5}+24{x}^{3}+48x$

Ethan Sanders

Step 1
$P\left(x\right)=3{x}^{5}+24{x}^{3}+48x$
To find all the zeroes of the polynomial. Let, P(x)=0
$0=3\left({x}^{5}+8{x}^{3}+16x\right)$
$0=3x\left({x}^{4}+8{x}^{2}+16\right)$
$0=3x\left({x}^{4}+4{x}^{2}+4{x}^{2}+16\right)$
$0=3x\left[{x}^{2}\left({x}^{2}+4\right)+4\left({x}^{2}+4\right)\right]$
$0=3x\left[\left({x}^{2}+4\right)\left({x}^{2}+4\right)\right]$
$0=3x\left({x}^{2}+4\right)$
Step 2
Therefore,
Zeroes of the polynomial are
x=0
${x}^{2}+4=0$ i.e. ${x}^{2}=-4$
i.e. $x=±\sqrt{-4}$
$=±\sqrt{\left({i}^{2}×{2}^{2}\right)}$
$=±2i$
i.e.
x=2i and
x=-2i
Factorization of the polynomial is
$3x{\left({x}^{2}+4\right)}^{2}$

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