Question

Find factors of $f\left(x\right)=2x^4+7x^3-4x^2-27x-18$ using synthetic division.

• A. $(x-2)$
• B. $(x+3)$
• C. $(x-3)$
• D. $(x+4)$

Answer 1

Answer: (A), (B)

$(x-2)$
$(x+3)$

Given, $f\left(x\right)=2x^4+{7x}^3-{4x}^2-27x-18$

Put $x=-1$

$f(-1)=2-7-4+27-18=0$

So $(x+1)$ is a factor of $f\left(x\right)$

Dividing $f\left(x\right)$ by $(x+1)$ we get

$p\left(x\right)=2x^3+5x^2-9x-18$

Put $x=2$

$p\left(2\right)=16+20-18-18=0$

So $(x-2)$ is a factor of $p\left(x\right)$

Dividing $p\left(x\right)$ by $(x-2)$ we get

$q\left(x\right)=2x^2+9x+9$

Factorising $q\left(x\right)$

$2x^2+9x+9=02x^2+6x+3x+9=02x(x+3)+3(x+3)=0(2x+3)(x+3)=0$

Now $f\left(x\right)=(x+1)p\left(x\right)q\left(x\right)$

$\Rightarrow f\left(x\right)=(x+1)(x-2)(2x+3)(x+3)$

$(x-2)$ and $(x+3)$ are factor of given polynomial.

So options $A$ and $B$ are correct.