How do you factor #2x^3+3x^2-16x-24#?

2 Answers
Alan P.
May 25, 2015

Given #2x^3+3x^2-16x-24#

Regroup as
#color(red)(2x^3-16x) + color(blue)(3x^2-24)#

Extract factor from each pair
#= color(red)((2x)(x^2-8)) + color(blue)((3)(x^2-8)#

Extract the #(x^2-8)# common factor
#=(2x+3)(x^2-8)#

If you are willing to employ irrational constants, this can be further factored (using the difference of squares) as
#=(2x-3)(x+2sqrt(2))(x-2sqrt(2))#

George C.
May 25, 2015

Notice that the ratio between the 1st and 2nd term is the same as that between the 3rd and 4th term. So grouping will work...

#2x^3+3x^2-16x-24 = (2x^3+3x^2)-(16x+24)#

#=x^2(2x+3)-8(2x+3)#

#=(x^2-8)(2x+3)#

#=(x-sqrt(8))(x+sqrt(8))(2x+3)#

#=(x-2sqrt(2))(x+2sqrt(2))(2x+3)#

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