Question
Factorize the following polynomials:
$4x^3 + 20x^2 + 33x + 18$ given that $2x + 3$ is a factor.
$4x^3 + 20x^2 + 33x + 18$ given that $2x + 3$ is a factor.
✓
Answer
Let f(x) =$ 4x^3 + 20x^2 + 33x + 18$ be the given polynomial.
Therefore, $(2x + 3)$ is a factor of the polynomial f(x).
Now,
f(x) = $2x^2(2x + 3) + 7x(2x + 3) + 6(2x + 3)$
$= (2x + 3)(2x^2 + 7x + 6)$
$= (2x + 3)(2x^2 +4x + 3x + 6)$
$= (2x + 3)(2x + 3)(x + 2)$
Hence$ (x + 2), (2x + 3)$ and $(2x + 3)$ are the factors of polynomial f(x).
Therefore, $(2x + 3)$ is a factor of the polynomial f(x).
Now,
f(x) = $2x^2(2x + 3) + 7x(2x + 3) + 6(2x + 3)$
$= (2x + 3)(2x^2 + 7x + 6)$
$= (2x + 3)(2x^2 +4x + 3x + 6)$
$= (2x + 3)(2x + 3)(x + 2)$
Hence$ (x + 2), (2x + 3)$ and $(2x + 3)$ are the factors of polynomial f(x).
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