How do you factor #y= x^5 - 16x^3 + 8x^2 - 128# ?

1 Answer
Jan 6, 2016

Answer:

Factor by grouping then using a couple of identities to find:

#x^5-16x^3+8x^2-128 = (x+2)(x^2-2x+4)(x-4)(x+4)#

Explanation:

The sum of cubes identity can be written:

#a^3+b^3=(a+b)(a^2-ab+b^2)#

The difference of squares identity can be written:

#a^2-b^2=(a-b)(a+b)#

Factor by grouping then use these two identities as follows:

#x^5-16x^3+8x^2-128#

#=(x^5-16x^3)+(8x^2-128)#

#=x^3(x^2-16)+8(x^2-16)#

#=(x^3+8)(x^2-16)#

#=(x^3+2^3)(x^2-4^2)#

#=(x+2)(x^2-(x)(2)+2^2)(x-4)(x+4)#

#=(x+2)(x^2-2x+4)(x-4)(x+4)#