Question

Determine which of these is a factor of the polynomial $P\left(x\right)=2x^4+x^3-26x^2+11x+12$ using synthetic division.

• A. $(x+1)$
• B. $(x+3)$
• C. $(x+4)$
• D. $(2x-1)$

Answer 1

The correct option is C $(x+4)$

Let $f\left(x\right)=2x^4+x^3-26x^2+11x+12$

We generally find the first root by hit and trial method

So put $x=1$

$f\left(1\right)=2+1-26+11+12=0$

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

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

$p\left(x\right)=2x^3+3x^2-23x-12$

Put $x=3$

$p\left(3\right)=54+27-69-12=0$

So $(x-3)$ is a factor of $p\left(x\right)$

Dividing $p\left(x\right)$ by $(x-3)$ we get

$q\left(x\right)=2x^2+9x+4$

Factorising $q\left(x\right)$

$q\left(x\right)=2x^2+8x+x+4q\left(x\right)=2x(x+4)+1(x+4)q\left(x\right)=(2x+1)(x+4)$

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

$f\left(x\right)=(x-1)(x-3)(2x+1)(x+4)$

$(x+4)$ is a factor,

So option $C$ is correct.