Remainder Theorem

Question

Use the Remainder Theorem to determine whether x + 13 is a factor of the polynomial

 

𝑓(𝑥) = 𝑥^4 + 12𝑥^3 − 9𝑥^2 + 48𝑥 − 52.

Answer 1

According to the Remainder Theorem, the remainder when f(x) is divided by x-c is f(c).

 

So, for the given function, the remainder when f(x) is divided by x+13 (or x-(-13)) is f(-13).

 

   f(-13) = (-13)⁴ +12(-13)³ - 9(-13)² + 48(-13) - 52 = 0

 

Since the remainder is 0, x+13 is a factor of f(x).  

Answer 2

 

-13 | 1   12   -9   48   -52

            -13   13  -52   52

        1    -1    4   -4      0

 

QED