Question

Using the Remainder Theorem, find the remainder obtained when $f\left(x\right)=5x^2+3x-1$ is divided by $g\left(x\right)=x-3$.

Answer 1

we know that If the value of the polynomial $g\left(x\right)$ becomes equal to 0 when we substitute $x=a$, then $a$ is a zero of the polynomial $g\left(x\right)$, and vice-versa.

Therefore, we have
$g\left(x\right)=x-3For,g\left(x\right)=0,\Rightarrow x-3=0\Rightarrow x=3$

Therefore, 3 is the zero of the polynomial $f\left(x\right)$.

Now to get the value of the dividend at the zero of the divisor, we will put x = zero of the divisor in $f\left(x\right)$.

Therefore,
$f\left(x\right)=5x^2+3x-1$ (given)

Putting $x=3$ we get,
$f\left(3\right)=5\times 3^2+3\times 3-1f\left(3\right)=45+9-1f\left(3\right)=53$

Hence, $f\left(3\right)=53$.

Therefore 53 is the remainder here.