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

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

$f(x)=2 x^{3}-9 x^{2}+x+13, g(x)=3-2 x$

Solution:

Here, $f(x)=2 x^{3}-9 x^{2}+x+13$

g(x) = 3 - 2x

To prove that g(x) is the factor of f(x),

we should show ⟹ f(3/2) = 0

here, 3 - 2x = 0

⟹ -2x = -3

⟹ 2x = 3

⟹ x = 3/2

Substitute the value of x in f(x)

$f(3 / 2)=2(3 / 2)^{3}-9(3 / 2)^{2}+(3 / 2)+13$

= 2(27/8) − 9(9/4) + 3/2 + 12

= (27/4) − (81/4) + 3/2 + 12

Taking L.C.M

$=\frac{21-81+6+48}{4}$

$=\frac{81-81}{4}$

= 0

Since, the result is 0 g(x) is the factor of f(x)