Question
Use factor theorem to factorise the following polynomials completely: $4x^3 + 4x^2 – 9x – 9$
✓
Answer
$f(x)=4 x^3+4 x^2-9 x-9$
Let $x=-1$, then
$f(-1)=4(-1)^3+4(-1)^2-9(-1)-9$
$=4(-1)+4(1)+9-9$
$=-4+4+9-9$
$=13-13$
$=0$
$\therefore(x+1)$ is a factor of $f(x)$
Now dividing $f(x)$ by $x+1$, we get
$ f(x)=4 x^3+4 x^2-9 x-9$
$=(x+1)\left(4 x^2-9\right)$
$=(x+1)\left\{(2 x)^2-(3)^2\right\}$
$=(x+1)(2 x+3)(2 x-3)$
Let $x=-1$, then
$f(-1)=4(-1)^3+4(-1)^2-9(-1)-9$
$=4(-1)+4(1)+9-9$
$=-4+4+9-9$
$=13-13$
$=0$
$\therefore(x+1)$ is a factor of $f(x)$
Now dividing $f(x)$ by $x+1$, we get
$ f(x)=4 x^3+4 x^2-9 x-9$
$=(x+1)\left(4 x^2-9\right)$
$=(x+1)\left\{(2 x)^2-(3)^2\right\}$
$=(x+1)(2 x+3)(2 x-3)$
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