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Given a polynomial and one of its factors, find the remaining factors of the polynomial. x^4 + 2x^3 – 8x – 16; x + 2

Sagot :

MrRoyal

Answer:

The other factors are: [tex](x-2)[/tex] and [tex](x^2+2x+4)[/tex]

Step-by-step explanation:

Given

Polynomial: [tex]x^4 + 2x^3 - 8x - 16[/tex]

Factor: [tex]x + 2[/tex]

Required

Find other factors

[tex]x^4 + 2x^3 - 8x - 16[/tex]

Group into two

[tex]x^4 + 2x^3 - 8x - 16 = (x^4 + 2x^3) - (8x + 16)[/tex]

Factorize:

[tex]x^4 + 2x^3 - 8x - 16 = x^3(x + 2) -8 (x + 2)[/tex]

Factor out common term

[tex]x^4 + 2x^3 - 8x - 16 = (x^3 -8)(x + 2)[/tex]

Rewrite [tex]x^3- 8[/tex] as [tex]x^3 + 2x^2 - 2x^2+ 4x - 4x - 8[/tex]

[tex]x^4 + 2x^3 - 8x - 16 = (x^3 + 2x^2 - 2x^2+ 4x - 4x - 8)(x + 2)[/tex]

Rearrange the terms

[tex]x^4 + 2x^3 - 8x - 16 = (x^3 + 2x^2 + 4x - 2x^2 - 4x - 8)(x + 2)[/tex]

Factorize

[tex]x^4 + 2x^3 - 8x - 16 = (x(x^2+2x+4)-2(x^2+2x+4))(x + 2)[/tex]

Factor out [tex](x^2+2x+4)[/tex]

[tex]x^4 + 2x^3 - 8x - 16 = (x-2)(x^2+2x+4)(x + 2)[/tex]

So, the other factors are: [tex](x-2)[/tex] and [tex](x^2+2x+4)[/tex]

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