Question

If the polynomials $a{z}^3+4z^2+3z-4$ and $z^3-4z+a$ leave the same remainder when divided by$z-3$, then find the value of $a$.

Answer 1

Step 1: State the given data and the remainder theorem

It is given that the polynomials $a{z}^3+4z^2+3z-4$ and $z^3-4z+a$ leave the same remainder when divided by $z-3$.

Let us say, $f\left(z\right)=a{z}^3+4z^2+3z-4$

And, $g\left(z\right)=z^3-4z+a$.

Also, the divisor $=z-3$

According to the remainder theorem,

"If a polynomial $p\left(x\right)$ is divided by the binomial $x-a$, then the remainder obtained is $p\left(a\right)$."

Step 2: Calculate the value of $a$

Since $f\left(z\right)$ and $g\left(z\right)$ leave the same remainder when divided by $z-3$.

Then, using the remainder theorem, the values of the two functions will be equal at $z=3$.

i.e., $f\left(3\right)=g\left(3\right)$

$\Rightarrow$ $a{\left(3\right)}^3+4{\left(3\right)}^2+3\left(3\right)-4={\left(3\right)}^3-4\left(3\right)+a$

$\Rightarrow$ $27a+36+9-4=27-12+a$

$\Rightarrow$ $27a+41=15+a$

$\Rightarrow$ $27a-a=15-41$

$\Rightarrow$ $26a=-26$

$\Rightarrow$ $a=-\frac{26}{26}$

$\Rightarrow$ $a=-1$

Hence, the required value of $a$ is $-1$.