Question

Is (x-5) a factor of the following polynomial? Write a short sentence to explain why or why not.

2x^3 - 10x^2 + x - 5

A. (x-5) is a factor because it evenly divides the polynomial.
B. (x-5) is not a factor because it does not evenly divide the polynomial.
C. The factorization depends on the value of x.
D. (x-5) is a factor, but it is not the only factor.

Answer 1

To determine if (x-5) is a factor of the polynomial 2x^3 - 10x^2 + x - 5, apply the Remainder Theorem by evaluating the polynomial at x=5. The result is 0, confirming that (x-5) is indeed a factor.

Step-by-step explanation:

To determine whether (x-5) is a factor of the polynomial 2x^3 - 10x^2 + x - 5, we can use the Remainder Theorem. This theorem states that if a polynomial f(x) is divided by (x-c), the remainder is f(c). Therefore, we can evaluate the polynomial at x=5. If the result is 0, then (x-5) is indeed a factor of the polynomial.

Let's evaluate: 2(5)^3 - 10(5)^2 + 5 - 5 equals 250 - 250 + 5 - 5, which simplifies to 0. Since the remainder is 0, (x-5) is a factor of the polynomial 2x^3 - 10x^2 + x - 5. Therefore, the correct answer is:

A. (x-5) is a factor because it evenly divides the polynomial.