In each of the following, 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
We have,
f(x) = x5+3x4-x3-3x2+5x+15 and g(x) = x+3
In order to find whether g (x) = x – (-3) is a factor of f (x) or not, it is sufficient to prove that f (-3) = 0
Now,
f(x) = x5+3x4-x3-3x2+5x+15
f (-3) = (-3)5 + 3 (-3)4 – (-3)3 – 3 (-3)2 + 5 (-3) + 15
= - 243 + 243 – (-27) – 3 (9) + 5 (-3) + 15
= - 243 + 243 + 27 – 27 – 15 + 15
= 0
Hence, g (x) is a factor of f (x).