In each of the following, use factor theorem to find whether polynomia

1.	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) = 2x3 - 9x2 + x +12, g(x) = 3 - 2x
Answer» We have, f(x) = 2x³ - 9x² + x +12 and g(x) = 3 - 2x In order to find g (x) = 3 – 2x = 2 \((x-\frac{3}{2})\) is a factor of f (x) or not, it is sufficient to prove that f \((\frac{3}{2})=0\) Now, f(x) = 2x³ - 9x² + x +12 f \((\frac{3}{2})=2(\frac{3}{2})^{3}-9(\frac{3}{2})^{2}+\frac{3}{2}+12\) \(=\frac{27}{4}-\frac{81}{4}+\frac{3}{2}+12\) = \(\frac{81-81}{4}\) = 0 Hence, g (x) is a factor of f (x).

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) = 2x3 - 9x2 + x +12, g(x) = 3 - 2x

Answer»

We have,

f(x) = 2x³ - 9x² + x +12 and g(x) = 3 - 2x

In order to find g (x) = 3 – 2x = 2 \((x-\frac{3}{2})\) is a factor of f (x) or not, it is sufficient to prove that f \((\frac{3}{2})=0\)

Now,

f(x) = 2x³ - 9x² + x +12

f \((\frac{3}{2})=2(\frac{3}{2})^{3}-9(\frac{3}{2})^{2}+\frac{3}{2}+12\)

\(=\frac{27}{4}-\frac{81}{4}+\frac{3}{2}+12\)

= \(\frac{81-81}{4}\)

= 0

Hence,

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