Question

if x^3 is divided by 2x^2-1 then the possible degree of quotient is

Answer 1

Polynomial division

Degree of a polynomial. We consider a polynomial of the form

$p(x)=a_{0}x^{n}+a_{1}x^{n-1}+...+a_{n-1}x+a_{n}$

where $a_{0},\:a_{1},\:...,\:a_{n}$ are given numbers (real or complex), $n$ is a non-negative integer and $x$ is a variable.

If $a_{0}\neq 0$, the polynomial $p(x)$ is said to be of degree $n$.

Division algorithm. Let $f(x)$ and $g(x)$ be two polynomials of degree $n$ and $m$ respectively and $n\geqslant m$. Then there exist two uniquely determined polynomials $q(x)$ and $r(x)$ satisfying

$\quad\quad f(x)=g(x)\:q(x)+r(x)$,

where the degree of $\color{red}{q(x)}$ is $\color{red}{n-m}$ and $r(x)$ is either a zero polynomial or the degree of $r(x)$ is less than $m$.

Solution.

Given,

$\quad$Dividend, $f(x)=x^{3}$

$\quad$Divisor, $g(x)=2x^{2}-1$

Here degree of $f(x)$ is $\color{red}{n=3}$

$\:$ and degree of $g(x)$ is $\color{red}{m=2}$.

Hence the degree of the quotient is

$\quad\quad n-m=3-2=\color{red}{1}$.