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Using factor theorem, factorize each of the following polynomial:

x3 + 6x2 + 11x + 6

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Let f (x) = x3 + 6x2 + 11x + 6 be the given polynomial.

The constant term in f (x) is 6 and factors of 6 are +1,+2,+3 and +6

Putting x = - 1 in f (x) we have, 

f (-1) = (-1)3 + 6 (-1)2 + 11 (-1) + 6 

= -1 + 6 – 11 + 6 

= 0 

Therefore, 

(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 cannot have more than three linear factors. 

Therefore, 

f (x) = k (x + 1) (x + 2) (x + 3) 

x3 + 6x2 + 11x + 6 = k (x + 1) (x + 2) (x + 3) 

Putting x = 0, on both sides we get, 

0 + 0 + 0 + 6 = k (0 + 1) (0 + 2) (0 + 3) 

6 = 6k 

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, 

x3 + 6x2 + 11x + 6 = (x + 1) (x + 2) (x + 3)

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