Question

In the following case, use the remainder theorem and find the remainder when $p\left(x\right)$ is divided by $g\left(x\right)$. $p\left(x\right)=4x^3-12x^2+14x-3g\left(x\right)=2x-1$

Answer 1

The Remainder Theorem states that when you divide a polynomial $p\left(x\right)$ by any factor $(x-a)$; which is not necessarily a factor of the polynomial; you will obtain a new smaller polynomial and a remainder, and this remainder is the value of $p\left(x\right)$ at $x=a$, that is $p\left(a\right)$.

Here, it is given that the polynomial $p\left(x\right)=4x^3-12x^2+14x-3$ and the factor is $g\left(x\right)=2x-1$, therefore, by remainder theorem, the remainder is $p\left(\frac{1}{2}\right)$ that is:

$p\left(\frac{1}{2}\right)=4{\left(\frac{1}{2}\right)}^3-12{\left(\frac{1}{2}\right)}^2+(14\times \frac{1}{2})-3=(4\times \frac{1}{8})-(12\times \frac{1}{4})+7-3=\frac{1}{2}-3+7-3=\frac{1}{2}+1=\frac{3}{2}$

Hence, the remainder is $r\left(x\right)=p\left(\frac{1}{2}\right)=\frac{3}{2}$.