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