How do you factor x^5 -4x^4 + 8x^3 - 14x^2 +15x -6 x5−4x4+8x3−14x2+15x−6?
1 Answer
Find factor
x^5-4x^4+8x^3-14x^2-14x^2+15x-6x5−4x4+8x3−14x2−14x2+15x−6
= (x-1)(x-1)(x-2)(x^2+3)=(x−1)(x−1)(x−2)(x2+3)
Explanation:
f(x) = x^5-4x^4+8x^3-14x^2-14x^2+15x-6f(x)=x5−4x4+8x3−14x2−14x2+15x−6
First notice that
So
f(x) = (x-1)(x^4-3x^3+5x^2-9x+6)f(x)=(x−1)(x4−3x3+5x2−9x+6)
Again, the remaining quartic factor is divisible by
x^4-3x^3+5x^2-9x+6 = (x-1)(x^3-2x^2+3x-6)x4−3x3+5x2−9x+6=(x−1)(x3−2x2+3x−6)
The remaining cubic factor is factorizable by grouping:
x^3-2x^2+3x-6 = (x^3-2x^2)+(3x-6) = x^2(x-2)+3(x-2)x3−2x2+3x−6=(x3−2x2)+(3x−6)=x2(x−2)+3(x−2)
= (x^2+3)(x-2)=(x2+3)(x−2)
The remaining quadratic factor has no linear factors with Real coefficients since
Putting this together we get:
x^5-4x^4+8x^3-14x^2-14x^2+15x-6 = (x-1)(x-1)(x-2)(x^2+3)
If you allow Complex coefficients then you can factor further using:
x^2+3 = (x-sqrt(3)i)(x+sqrt(3)i)
So:
x^5-4x^4+8x^3-14x^2-14x^2+15x-6
= (x-1)(x-1)(x-2)(x-sqrt(3)i)(x+sqrt(3)i)
