Using factor theorem, factorize each of the following polynomial:x3 +

1.	Using factor theorem, factorize each of the following polynomial:x3 + 6x2 + 11x + 6
Answer» 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)

Using factor theorem, factorize each of the following polynomial:x3 + 6x2 + 11x + 6

Answer»

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)