How do you use the factor theorem to determine whether a-1 is a factor of $a^3 – 2a^2 + a – 2$ ?

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Answer 1 · 2018-03-10T08:12:57

$(a-1)$ is not a factor of $a^3-2a^2+a-2$

According to factor theorem if $x-a$ is a factor of polynomial function $f(x)$ , then $f(a)=0$ . Now let $(x-1)$ be a factor of the function

$f(x)=a_0x^n+a_1x^(n-1)+a_2x^(n-2)+.......+a_n$

then $f(1)=a_0+a_1+a_2+.......+a_n=0$

This means that if $(a-1)$ is a factor of polynomial function $a^3-2a^2+a-2$ , then sum of its coefficients must be zero.

Here as $1-2+1-2=-2$ and sum of coefficients is not zero,

$(a-1)$ is not a factor of $a^3-2a^2+a-2$

How do you use the factor theorem to determine whether a-1 is a factor of a^3 – 2a^2 + a – 2a^3 – 2a^2 + a – 2?