Using factor theorem, factorize each of the following polynomial:
x³+6x²+11x+6
Answers
Using factor theorem, the factors of x³+6x²+11x+6 are (x+1), (x+2) and (x+3)
Let,
f(x) = x^3 + 6x^2 + 11x + 6
now let us take, x = -1
f(-1) = (-1)^3 + 6(-1)^2 + 11(-1) + 6 = 0
∴ (x+1) is a factor of f(x)
In order to find remaining factors, we need to divide the given polynomial using the above obtained factor (x+1)
(x+1) x^3 + 6x^2 + 11x + 6 (x^2+5x+6)
x^3+x^2
-------------
5x^2+11x+6
5x^2+5x
-------------
6x+6
6x+6
----------
0
Another factor obtained is x^2+5x+6
Let us further obtain the factors of this quadratic equation
x^2+5x+6
= x^2+2x+3x+6
= x(x+2) + 3(x+2)
= (x+2) (x+3)
⇒ The factors of the polynomial x^3 + 6x^2 + 11x + 6 = (x+1) (x+2) (x+3)