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