How do you factor x^3+9x^2+24x+16 if you know that x+4 is a factor?

2 Answers
Dec 17, 2017

x^3+9x^2+24x+16 = (x+4)(x+4)(x+1)x3+9x2+24x+16=(x+4)(x+4)(x+1)

Explanation:

x^3+9x^2+24x+16 = (x+4)(x^2+5x+4)x3+9x2+24x+16=(x+4)(x2+5x+4)

color(white)(x^3+9x^2+24x+16) = (x+4)(x+4)(x+1)x3+9x2+24x+16=(x+4)(x+4)(x+1)

A little more slowly...

x^3+9x^2+24x+16 = x^3+4x^2+5x^2+20x+4x+16x3+9x2+24x+16=x3+4x2+5x2+20x+4x+16

color(white)(x^3+9x^2+24x+16) = x^2(x+4)+5x(x+4)+4(x+4)x3+9x2+24x+16=x2(x+4)+5x(x+4)+4(x+4)

color(white)(x^3+9x^2+24x+16) = (x^2+5x+4)(x+4)x3+9x2+24x+16=(x2+5x+4)(x+4)

color(white)(x^3+9x^2+24x+16) = (x^2+4x+x+4)(x+4)x3+9x2+24x+16=(x2+4x+x+4)(x+4)

color(white)(x^3+9x^2+24x+16) = (x(x+4)+1(x+4))(x+4)x3+9x2+24x+16=(x(x+4)+1(x+4))(x+4)

color(white)(x^3+9x^2+24x+16) = (x+1)(x+4)(x+4)x3+9x2+24x+16=(x+1)(x+4)(x+4)

Dec 17, 2017

(x+4)^2(x+1)(x+4)2(x+1)

Explanation:

color(red)(x^2)(x+4)color(magenta)(-4x^2)+9x^2+24x+16x2(x+4)4x2+9x2+24x+16

=color(red)(x^2)(x+4)color(red)(+5x)(x+4)color(magenta)(-20x)+24x+16=x2(x+4)+5x(x+4)20x+24x+16

=color(red)(x^2)(x+4)color(red)(+5x)(x+4)color(red)(+4)(x+4)cancel(color(magenta)(-16))cancel(+16)

=(x+4)(color(red)(x^2+5x+4))

=(x+4)(x+4)(x+1)

rArrx^3+9x^2+24x+16=(x+4)^2(x+1)