Use factor theorem to find whether polynomial g(x) is a factor of polynomial f(x), or not:
$f(x)=x^{3}-6 x^{2}+11 x-6, g(x)=x^{2}-3 x+2$
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)
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