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

2 Answers
George C.
Dec 17, 2017

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

Explanation:

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

#color(white)(x^3+9x^2+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+16#

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

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

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

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

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

Jim G.
Dec 17, 2017

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

Explanation:

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

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

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