# A polynomial P is given (a) Factor P into linear

A polynomial P is given (a) Factor P into quadratic factors with real coefficients that are linear and irreducible. (b) Consider only linear components with complex coefficients for P..
$P\left(x\right)={x}^{5}-16x$

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a) To find: The polynomial P(x) is factored by linear, irreducible, quadratic factors with real coefficients.
Given information:
The polynomial P(x) is,
$P\left(x\right)={x}^{5}-16x$
Concept used:
The product of linear and irreducible quadratic factors can be factored into any polynomial with real coefficients.
Calculation:
The given polynomial P(x) is,
$P\left(x\right)={x}^{5}-16x$
Rewrite the above polynomial as,
$P\left(x\right)=x\left({x}^{4}-16\right)$
$=x\left({\left({x}^{2}\right)}^{2}-{4}^{2}\right)$
Use identity ${a}^{2}-{b}^{2}=\left(a-b\right)\left(a+b\right)$ to factor the above equation as,
$P\left(x\right)=x\left({x}^{2}-4\right)\left({x}^{2}+4\right)$
$=x\left({x}^{2}-{2}^{2}\right)\left({x}^{2}+4\right)$
$=x\left(x-2\right)\left(x+2\right)\left({x}^{2}+4\right)$
The factors x, $\left(x-2\right)$ and $\left(x+2\right)$ are linear factors.
The factor $\left({x}^{2}+4\right)$ is irreducible, since it has no real zeros.
Conclusion:
Thus, the factored form of the polynomial P(x) that has linear and irreducible quadratic factors is $x\left(x-2\right)\left(x+2\right)\left({x}^{2}+4\right)$
b) To find: The factors of the polynomial P(x) that has linear factors with complex coefficients.
Given: The polynomial P(x) is,
$P\left(x\right)={x}^{5}-16x$
Calculation:
From part (a) the factored form of the polynomial P(x) is,
$P\left(x\right)=x\left(x-2\right)\left(x+2\right)\left({x}^{2}+4\right)$
Now, factor the remaining quadratic factor to obtain the complete factorization as,
$P\left(x\right)=x\left(x-2\right)\left(x+2\right)\left({x}^{2}+4\right)$
$=x\left(x-2\right)\left(x+2\right)\left({x}^{2}-{\left(2i\right)}^{2}\right)$
$=x\left(x-2\right)\left(x+2\right)\left(x-2i\right)\left(x+2i\right)$
The above factors are linear factors with complex coefficients.
Conclusion:
Thus, the factored form of the polynomial P(x) that has linear factors is $x\left(x-2\right)\left(x+2\right)\left(x-2i\right)\left(x+2i\right)$.