Question

In the following case, use the remainder theorem and find the remainder when $p\left(x\right)$ is divided by $g\left(x\right)$. $p\left(x\right)=4x^3-10x^2+12x-3g\left(x\right)=x+1$

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 $p\left(x\right)=4x^3-10x^2+12x-3$ and the factor is $g\left(x\right)=x+1$, therefore, by remainder theorem, the remainder is $p(-1)$ that is:

$p(-1)=(4\times (-1{)}^3)-(10\times (-1{)}^2)+(12\times (-1\left)\right)-3=(4\times (-1\left)\right)-(10\times 1)-12-3$
$=-4-10-12-3=-29$

Hence, the remainder is $r\left(x\right)=p(-1)=-29$.