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Question

Using factor theorem, factorize the polynomial x36x2+11x6.

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Solution

Let f(x) = x36x2+11x6
The constant term in f(x) is equal to -6 and factors of -6 are ±1,±2,±3,±6.
Putting x = 1 in f(x), we have
f(1)=136×12+11×16
= 1 - 6 + 11 - 6 = 0
(x - 1) is a factor of f(x)
Similarly, x - 2 and x - 3 are factors of f(x)
Since f(x) is a polynomial of degree 3. So, it can not have more than three linear factors.
Let f(x) = k (x - 1) (x - 2) (x - 3). Then,
x36x2+11x6=k(x1)(x2)(x3)
Putting x = 0 on both sides, we get
-6 = k (0 - 1) (0 - 2) (0 - 3)
-6 = -6 k k = 1
Putting k = 1 in f(x) = k (x - 1) (x - 2) (x - 3), we get
f(x) = (x - 1) (x - 2) (x - 3)
Hence, x36x2+11x6=(x1)(x2)(x3)

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