Question

Drag each factor to the correct location on the image. If p(1) = 3, p(-4) = 8, p(5) = 0, p(7) = 9, p(-10) = 1, and p(-12) = 0, determine which expressions are factors of the polynomial p(x).

Answer 1

Factor: x-5 & x+12

Not a Factor: x-1, x+4, x-7, x+10

Explanation:

The remainder theorem states that for a polynomial p(x) and a number a, the remainder on division of p(x) by x − a is p(a). So, p(a) = 0 if and only if x − a is a factor of p(x).

Since, p(5) = 0 and p(-12) = 0, the expressions (x − 5) and (x + 12) are factors of the polynomial p(x).

Answer 2

Explanation:

Recall that when we are factoring a polynomial, we are implicitly finding its roots. By finding a root, we are looking for a value of x (say c) such that p(c) =0. When this happens, our polynomial has the the polynomial (x-c) as a factor.

We are given the values of p at some values of x. We notice that p(5) = 0 and p(-12) =0. So, this means that our polynomial has as a factor the polynomials (x-5) and (x+12).