Question

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

Answer 1

Hint: We have been given a polynomial expression $p\left(x\right)$ and we have to determine whether another expression $x-a$ is a factor of the given polynomial or not. For that, we use remainder theorem. According to that, if $x-a$ is a factor of $p\left(x\right)$ then $p\left(a\right)=0$ . In this question, first , we equate the $x-a$ to zero and determines the value of $x=a$ . Now we substitute this value of $x$ in the given polynomial $p\left(x\right)$ and check whether the value of the polynomial is $0$ or not. If the obtained value is equal to zero then the expression $x-a$ is the factor of $p\left(x\right)$ otherwise not.

Complete step-by-step solution:
Step1: Given polynomial is $4x^4-2x^3+3x^2-2x+1$ and we have to determine whether $x-1$ is a factor of the above given polynomial or not. For that, we equate $x-1=0$ so we get the value of $x=1$
Step2: Now we use remainder theorem to determine whether $x-1$ is a factor of $4x^4-2x^3+3x^2-2x+1$ or not. For that, we substitute the value of $x=1$ in the given polynomial, we get
$4{\left(1\right)}^4-2{\left(1\right)}^3+3{\left(1\right)}^2-2\left(1\right)+1$
On simplification we get
$$\begin{gathered} \Rightarrow 4\times 1-2\times 1+3\times 1-2\times 1+1 \\ \Rightarrow 4-2+3-2+1 \\ \Rightarrow 4 \end{gathered}$$
Step3: Since the obtained value of the given polynomial $4x^4-2x^3+3x^2-2x+1$ at $x=1$ is not equal to zero.

So $x-1$ is not the factor of the polynomial $4x^4-2x^3+3x^2-2x+1$.

Note: While evaluating the power of a number remember that the odd power of a negative number results in a negative number, while the even power of a negative number results in a positive number.
In the case of a positive number, the even power or odd power in both cases, results in a positive number.