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..

$P\left(x\right)={x}^{5}-16x$

Tabansi
2021-08-14
Answered

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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falhiblesw

Answered 2021-08-15
Author has **97** answers

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:

Linear and Quadratic Factors Theorem:

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,

$P\left(x\right)=x({x}^{4}-16)$

$=x({\left({x}^{2}\right)}^{2}-{4}^{2})$

Use identity ${a}^{2}-{b}^{2}=(a-b)(a+b)$ to factor the above equation as,

$P\left(x\right)=x({x}^{2}-4)({x}^{2}+4)$

$=x({x}^{2}-{2}^{2})({x}^{2}+4)$

$=x(x-2)(x+2)({x}^{2}+4)$

The factors x, $(x-2)$ and $(x+2)$ are linear factors.

The factor $({x}^{2}+4)$ 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 $x(x-2)(x+2)({x}^{2}+4)$

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,

$P\left(x\right)=x(x-2)(x+2)({x}^{2}+4)$

Now, factor the remaining quadratic factor to obtain the complete factorization as,

$P\left(x\right)=x(x-2)(x+2)({x}^{2}+4)$

$=x(x-2)(x+2)({x}^{2}-{\left(2i\right)}^{2})$

$=x(x-2)(x+2)(x-2i)(x+2i)$

The above factors are linear factors with complex coefficients.

Conclusion:

Thus, the factored form of the polynomial P(x) that has linear factors is $x(x-2)(x+2)(x-2i)(x+2i)$.

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