Question:
Use factor theorem to find whether polynomial g(x) is a factor of polynomial f(x), or not:
$f(x)=x^{3}-6 x^{2}-19 x+84, g(x)=x-7$
Solution:
Here, $f(x)=x^{3}-6 x^{2}-19 x+84$
g(x) = x - 7
To prove that g(x) is the factor of f(x),
we should show ⟹ f(7) = 0
here, x - 7 = 0
⟹ x = 7
Substitute the value of x in f(x)
$f(7)=7^{3}-6(7)^{2}-19(7)+84$
= 343 - (6 * 49) - (19 * 7) + 84
= 342 - 294 - 133 + 84
= 427 - 427
= 0
Since, the result is 0 g(x) is the factor of f(x)