f(x) = x^5 + 3x^4 – x^3 – 3x^2 + 5x + 15, g(x) = x + 3

Question

Use factor theorem to find whether polynomial g(x) is a factor of polynomial f(x) or, not:

f(x) = x⁵ + 3x⁴ – x³ – 3x² + 5x + 15, g(x) = x + 3

Answer 1

If g(x) is a factor of f(x), then the remainder will be zero that is g(x) = 0. 

g(x) = x + 3 = 0, then x = -3

Remainder = f(-3)

Now, 

f(-3) = (-3)⁵ + 3(-3)⁴ – (-3)³ – 3(-3)² + 5(-3) + 15 

= -243 + 3 x 81 -(-27)-3 x 9 + 5(-3) + 15

= -243 + 243 + 27 - 27 - 15 + 15 

= 0

Therefore, g(x) is a factor of f(x).