(50 points) Show that x+2 is a factor of p(x)=x^4+2x^3+4x^2+x-14. Then write p(x) as the product of x+2 and another factor. Please show work.

Question

(50 points) Show that x+2 is a factor of p(x)=
`x^4+2x^3+4x^2+x-14`. Then write p(x) as the product of x+2 and another factor. Please show work.

Answer 1

Let p(x)=x

3

−4x

2

−2x+20

By factor theorem, (x+2) is a factor of p(x) if p(−2)=0.

∴ It is sufficient to show that (x+2) is a factor of p(x)

Now, p(x)=x

3

−4x

2

−2x+20

∴p(−2)=(−2)

3

−4(−2)

2

−2(−2)+20=−8−16+4+20=0

∴(x+2) is a factor of p(x)=x

3

−4x

2

−2x+20