Question

Which theorem equates the value of a polynomial at a given value ,k, to the value of the synthetic division of the polynomial by x-k?

Answer 1

It's the remainder theorem

Answer 2

The remainder theorem

Explanation:

The remainder theorem states that given a polynomial f(x) and a linear factor x - k as in the question, then if f(x) is divided by x - k, then remainder is obtained at the value of x - k = 0 ⇒ x = k. That is, the remainder is f(k) when x = k.

For example, if we have a polynomial f(x) = x³ + 3x² -2x + 1 and a linear factor x - 1. The remainder theorem states that the remainder when f(x) is divided by x - 1 is obtained by equating the factor to zero and finding the value of x.

So, x - 1 = 0 ⇒ x = 1

Substituting x = 1 into f(x), we have

f(x) = x³ + 3x² -2x + 1

f(1) = 1³ + 3(1)² -2(1) + 1

= 1 + 3 - 2 + 1

= 3.

So, the remainder is 3.