Question
Using factor theorem, factorize the following polynomials:
x3 + 6x2 + 11x + 6
x3 + 6x2 + 11x + 6
✓
Answer
Let x = 1
$\text{f(1)}=1^3+6(2)^2+11(1)+6\neq0$
Let x = -1
f(-1) = (-1)3 + 6(-1)2 + 11(-1) + 6 = 12 - 12 = 0
$\therefore$ x = -1 is a solution
⇒ x + 1 = 0
i. e (x + 1) is a factor of f(x)
By division algorithm
x3 + 6x2 + 11x + 6 = (x + 1)(x2 + 5x + 6)
= (x + 1)(x2 + 2x + 3x + 6)
= (x + 1)(x(x + 2) + 3(x + 2))
= (x + 1)(x + 2)(x + 3)
$\therefore$ x3 + 6x2 + 11x + 6 = (x + 1)(x + 2)(x + 3)
$\text{f(1)}=1^3+6(2)^2+11(1)+6\neq0$
Let x = -1
f(-1) = (-1)3 + 6(-1)2 + 11(-1) + 6 = 12 - 12 = 0
$\therefore$ x = -1 is a solution
⇒ x + 1 = 0
i. e (x + 1) is a factor of f(x)
By division algorithm
x3 + 6x2 + 11x + 6 = (x + 1)(x2 + 5x + 6)
= (x + 1)(x2 + 2x + 3x + 6)
= (x + 1)(x(x + 2) + 3(x + 2))
= (x + 1)(x + 2)(x + 3)
$\therefore$ x3 + 6x2 + 11x + 6 = (x + 1)(x + 2)(x + 3)
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