Question

Check whether the polynomial $p\left(y\right)=4y^3+4y^2-y-1$ is a multiple of $(2y+1)$.

Answer 1

To check that $2y+1$ is a multiple $p\left(y\right)$ or not, it will be sufficient to check whether $2y+1$ is a factor of $p\left(y\right)$ or not.

For that,

$2y+1=0$

$\Rightarrow y=-\frac{1}{2}$

Now, substitute $y=-\frac{1}{2}$ in the polynomial $p\left(y\right),$

$p(-\frac{1}{2})=4{(-\frac{1}{2})}^3+4{(-\frac{1}{2})}^2-(-\frac{1}{2})-1$

$\Rightarrow p(-\frac{1}{2})=4\times (-\frac{1}{8})+(4\times \frac{1}{4})+\frac{1}{2}-1$

$\Rightarrow p(-\frac{1}{2})=-\frac{1}{2}+1+\frac{1}{2}-1$

$\Rightarrow p(-\frac{1}{2})=0$

Here, the remainder is zero $0$, when the polynomial $p\left(y\right)=4y^3+4y^2-y-1$ is divided by $2y+1$.

So, by using the factor theorem now we can say that $p\left(y\right)$ is a multiple of $2y+1.$