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

Feb 17, 2017

See explanation.

Explanation:

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

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

$P \left(1\right) = 4 \cdot {1}^{4} - 2 \cdot {1}^{3} + 3 \cdot {1}^{2} - 2 \cdot 1 + 1 = 4 - 2 + 3 - 2 + 1$

$P \left(1\right) = 4$

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

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