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:

$f(x)=x^{3}-6 x^{2}+11 x-6, g(x)=x^{2}-3 x+2$

Solution:

Here, $f(x)=x^{3}-6 x^{2}+11 x-6$

$g(x)=x^{2}-3 x+2$

First we need to find the factors of $x^{2}-3 x+2$

$\Rightarrow x^{2}-2 x-x+2$

⟹ x(x - 2) -1(x - 2)

⟹ (x - 1) and (x - 2) are the factors

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

The results of f(1) and f(2) should be zero

Let, x – 1 = 0

x = 1

substitute the value of x in f(x)

$f(1)=1^{3}-6(1)^{2}+11(1)-6$

= 1 – 6 + 11 – 6

= 12 – 12

= 0

Let, x – 2 = 0

x = 2

substitute the value of x in f(x)

$f(2)=2^{3}-6(2)^{2}+11(2)-6$

= 8 – (6 * 4) + 22 – 6

= 8 – 24 + 22 - 6

= 30 – 30

= 0

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