How do you determine whether x-1 is a factor of the polynomial 4x^4-2x^3+3x^2-2x+14x4−2x3+3x2−2x+1?

1 Answer
Daniel L. · Shwetank Mauria
Feb 17, 2017

See explanation.

Explanation:

According to The Remainder Theorem (x-a)(x−a) is a factor of a polynomial P(x)P(x) if and only if P(a)=0P(a)=0.

So to check if (x-1)(x−1) is a factor of P(x)=4x^4-2x^3+3x^2-2x+1P(x)=4x4−2x3+3x2−2x+1 you have to check if P(1)=0P(1)=0

P(1)=4*1^4-2*1^3+3*1^2-2*1+1=4-2+3-2+1P(1)=4⋅14−2⋅13+3⋅12−2⋅1+1=4−2+3−2+1

P(1)=4P(1)=4

P(1)P(1) is not zero, so (x-1)(x−1) is not the factor of P(x)P(x)

In fact P(1)=4P(1)=4 means that the remainder when 4x^4-2x^3+3x^2-2x+14x4−2x3+3x2−2x+1 is divided by x-1x−1 is 44

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