use factor theorem to find whether polynomial g(x) is a factor of polynomial f(x) or, not: f(x) = 3x^4+17x^3 + 9x^2 - 7x -10, g(x) = x + 5

Question

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) = 3x⁴+17x³ + 9x² - 7x -10, g(x) = x + 5

Answer 1

We have,

f(x) = 3x⁴ + 17x³ + 9x² - 7x - 10 and g(x) = x + 5'

In order to find whether the polynomials g (x) = x – (-5) is a factor of f (x) or not, it is sufficient to show that f (-5) = 0

Now,

f(x) = 3x⁴ + 17x³ + 9x² - 7x - 10

f (-5) = 3 (-5)⁴ + 17 (-5)³ + 9 (-5)2 – 7 (-5) – 10

= 3 * 625 + 17 * (-125) + 9 * 25 + 35 – 10

= 1875 – 2125 + 225 + 35 – 10

= 0

Hence, 

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