Question

Using factor theorem, factorize the following polynomials

$x^3+6x^2$ + 11x + 6

Answer 1

f(x) = $x^3+6x^2$ + 11x + 6

The factors of constant term 6 are $\pm 1,\pm 2,\pm 3,and\pm 6,$

Let x = - 1, the

f(-1) $=(-1{)}^3+6(-1{)}^2$ + 11(-1)+6

= - 1 + 6 - 11 + 6 = 12 - 12 = 0

$\therefore$ f(-1) = 0

$\therefore$ x + 1 is the factor of f(x)

Let x = -2, then

f(-2) $=(-2{)}^3+6(-2{)}^2$ + 11(-2)+6

=- 8 + 24 - 22 + 6 = 30 - 30 = 0

$\therefore$ f(-2)=0

$\therefore$ x + 2 is a factor of f(x)

Let x = - 3, then

f(-3) $=(-3{)}^3+6(-3{)}^2$ + 11(-3)+6

= - 27 + 54 - 33 + 6 = 60 - 60 = 0

$\therefore$ f(-3) = 0

$\therefore$ x+ 3 is a factor of f(x)

$\therefore$ f(x) = (x+1) (x+2) (x+3)