Answer to Question #243129 in Algebra for enKay

Question

A)

The polynomial g(X)=3x^3-2x^2-12x+8

(I) using the factor theorem, show that (X+2) is a factor of g(X)

(II) factorize g(X) completely

Answer 1

(i)

$$x+2=0=>x=-2$$

$$g(-2)=3(-2)^3-2(-2)^2-12(-2)+8$$

$$=-24-8+24+8=0$$

Since $g(-2)=0,$ then by thefactor theorem $(x+2)$ is a factor of $g(x).$

(ii)

$$g(x)=3x^3-2x^2-12x+8$$

$$=3x^2(x+2)-8x(x+2)+4(x+2)$$

$$=(x+2)(3x^2-8x+4)$$

$$=(x+2)(3x(x-2)-2(x-2))$$

$$=(x+2)(x-2)(3x-2)$$