Question
Factorize the following polynomials:
$x^3+13 x^2+31 x-45$ given that $x+9$ is a factor.
$x^3+13 x^2+31 x-45$ given that $x+9$ is a factor.
✓
Answer
Let $f(x)=x^3+13 x^2+31 x-45$ be the given polynomial.
Therefore, $(x+9)$ is a factor of the polynomial $f(x)$.
Now,
$f(x)=x^2(x+9)+4 x(x+9)-5(x+9)$
$=(x+9)\left(x^2+4 x-5\right)$
$=(x+9)\left(x^2+5 x-x-5\right)$
$=(x+9)(x-1)(x+5)$
Hence $(x-1),(x+5)$ and $(x+9)$ are the factors of polynomial $f(x)$.
Therefore, $(x+9)$ is a factor of the polynomial $f(x)$.
Now,
$f(x)=x^2(x+9)+4 x(x+9)-5(x+9)$
$=(x+9)\left(x^2+4 x-5\right)$
$=(x+9)\left(x^2+5 x-x-5\right)$
$=(x+9)(x-1)(x+5)$
Hence $(x-1),(x+5)$ and $(x+9)$ are the factors of polynomial $f(x)$.
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