Question

In the following case, use the remainder theorem and find the remainder when $p\left(x\right)$ is divided by $g\left(x\right)$.
$p\left(x\right)=7x^3-x^2+2x-1g\left(x\right)=1-2x$

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 $p\left(x\right)=7x^3-x^2+2x-1$ and the factor is $g\left(x\right)=1-2x$, therefore, by remainder theorem, the remainder is $p\left(\frac{1}{2}\right)$ that is:

$p\left(\frac{1}{2}\right)=7{\left(\frac{1}{2}\right)}^3-{\left(\frac{1}{2}\right)}^2+(2\times \frac{1}{2})-1=(7\times \frac{1}{8})-\frac{1}{4}+1-1=\frac{7}{8}-\frac{1}{4}=\frac{7-2}{8}=\frac{5}{8}$

Hence, the remainder is $r\left(x\right)=p\left(\frac{1}{2}\right)=\frac{5}{8}$.