Question

how could you determine if x - 2 is a factor of (2x^{4}-5x^{2}+x - 2)?
use synthetic division and see if the remainder is zero
ask the person sitting next to me
use synthetic division and see if the quotient is even
flip a coin

Explanation:

Step1: Recall Factor Theorem

By the Factor Theorem, a polynomial $f(x)$ has a factor $(x-a)$ if and only if $f(a)=0$, which is equivalent to a remainder of 0 when $f(x)$ is divided by $(x-a)$.

Step2: Apply to given polynomial

For $f(x)=2x^3-5x^2+x-2$ and divisor $(x-2)$, use synthetic division to divide $f(x)$ by $(x-2)$. If the remainder is 0, $(x-2)$ is a factor.

Step3: Eliminate incorrect options

• "Ask the person sitting next to me" is not a mathematical method.
• "Use synthetic division and see if the quotient is even" is irrelevant, as the quotient's parity does not determine a factor.
• "Flip a coin" is a random choice, not a valid mathematical test.

Answer:

A. Use synthetic division and see if the remainder is zero