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