How do you factor given that f(6)=0f(6)=0 and f(x)=x^3-3x^2-16x-12f(x)=x33x216x12?

1 Answer
Sep 4, 2017

f(x)=(x-6)(x+1)(x+2)f(x)=(x6)(x+1)(x+2)

Explanation:

"using the "color(blue)"factor theorem"using the factor theorem

•color(white)(x)f(a)=0" if and only if "(x-a)" is a factor of "f(x)xf(a)=0 if and only if (xa) is a factor of f(x)

f(6)=0rArr(x-6)" is a factor of "f(x)f(6)=0(x6) is a factor of f(x)

f(x)=color(red)(x^2)(x-6)color(magenta)(+6x^2)-3x^2-16x-12f(x)=x2(x6)+6x23x216x12

color(white)(f(x))=color(red)(x^2)(x-6)color(red)(+3x)(x-6)color(magenta)(+18x)-16x-12f(x)=x2(x6)+3x(x6)+18x16x12

color(white)(f(x))=color(red)(x^2)(x-6)color(red)(+3x)(x-6)color(red)(+2)(x-6)color(magenta)(+12)-12f(x)=x2(x6)+3x(x6)+2(x6)+1212

color(white)(f(x))=color(red)(x^2)(x-6)color(red)(+3x)(x-6)color(red)(+2)(x-6)+0f(x)=x2(x6)+3x(x6)+2(x6)+0

rArrf(x)=(x-6)(color(red)(x^2+3x+2))f(x)=(x6)(x2+3x+2)

color(white)(rArrf(x))=(x-6)(x+1)(x+2)f(x)=(x6)(x+1)(x+2)