How do you determine whether x-1 is a factor of the polynomial x^3-3x^2+4x-2x3−3x2+4x−2?

1 Answer
sente
Jan 2, 2017

Observe that x^3-3x^2+4x-2x3−3x2+4x−2 evaluates to 00 at x=1x=1 and conclude that x-1x−1 is a factor of x^3-3x^2+4x-2x3−3x2+4x−2.

Explanation:

In general, given a polynomial P(x)P(x), we have that x-ax−a is a factor of P(x)P(x) if and only if P(a) = 0P(a)=0. To test if x-1x−1 is a factor of x^3-3x^2+4x-2x3−3x2+4x−2, then, we can evaluate x^3-3x^2+4x-2x3−3x2+4x−2 at 11:

1^3-3(1)^2+4(1)-2 = 1 - 3 + 4 - 2 = 013−3(1)2+4(1)−2=1−3+4−2=0

Thus x-1x−1 is a factor of x^3-3x^2+4x-2x3−3x2+4x−2.

If we want to see how it factors out, we can use polynomial long division to find that

x^3-3x^2+4x-2 = (x-1)(x^2-2x+2)x3−3x2+4x−2=(x−1)(x2−2x+2)

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