Question

Prove by factor theorem that,
ii) $(2x+1)\text{is a factor of}4x^3+12x^2+7x+1.$

Answer 1

To prove: $(2x+1)$ is a factor of $4x^3+12x^2+7x+1$.
From Factor theorem, we know that a polynomial $f\left(x\right)$ has a factor $(x–a)$ if and only if $f\left(a\right)=0$.
$2x+1=0\Rightarrow x=-\frac{1}{2}$
So, $a=-\frac{1}{2}$
$f\left(a\right)=f(-\frac{1}{2})$
$=4{(-\frac{1}{2})}^3+12{(-\frac{1}{2})}^2+7(-\frac{1}{2})+1$
$=4(-\frac{1}{8})+12\left(\frac{1}{4}\right)+7(-\frac{1}{2})+1$
$=(-\frac{1}{2})+3-\left(\frac{7}{2}\right)+1$
$=-4+4$
$=0$
$\therefore f\left(a\right)=0$
Hence,$(2x+1)\text{is a factor of}4x^3+12x^2+7x+1.$