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

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Answer 1 · 2017-02-17T13:03:38

See explanation.

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

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

$P(1)=4*1^4-2*1^3+3*1^2-2*1+1=4-2+3-2+1$

$P(1)=4$

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

In fact $P(1)=4$ means that the remainder when $4x^4-2x^3+3x^2-2x+1$ is divided by $x-1$ is $4$

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