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

1 Answer
sente
Jan 2, 2017

Observe that #x^3-3x^2+4x-2# evaluates to #0# at #x=1# and conclude that #x-1# is a factor of #x^3-3x^2+4x-2#.

Explanation:

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

#1^3-3(1)^2+4(1)-2 = 1 - 3 + 4 - 2 = 0#

Thus #x-1# is a factor of #x^3-3x^2+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)#

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