Using remainder theorem, find the remainder when: $f(x)=x^{2}+2ax+3a^{2},\ g( x)=x+a$.
Given: $f(x)=x^{2}+2ax+3a^{2},\ g( x)=x+a$.
To do: To find the remainder by using remainder theorem when $f( x)$ is divided by $g( x)$.
Solution:
As given, $f(x)=x^{2}+2ax+3a^{2}$ and $g( x)=x+a$.
Let $g( x)=x+a=0$
$\Rightarrow x=-a$, On substituting this value in $f( x)$.
$f( -a)=( -a)^2+2a( -a)+3a^2$
$\Rightarrow f( -a)=a^2-2a^2+3a^2$
$\Rightarrow f( -a)=2a^2$
Thus, the remainder is $2a^2$.
Related Articles
- Using remainder theorem, find the remainder when $f( x)$ is divided by $g( x)$:$f( x)=4 x^{3}-12 x^{2}+11 x-3,\ g( x)=x+\frac{1}{2}$.
- Use Remainder theorem to find the remainder when \( f(x) \) is divided by \( g(x) \) in the following $f(x)=x^{2}-5 x+7, g(x)=x+3$.
- Find the remainder when $x^3+ x^2 + x + 1$ is divided by $x - \frac{1}{2}$ using remainder theorem.
- In each of the following, using the remainder Theorem, find the remainder when $f(x)$ is divided by $g(x)$ and verify the result by actual division.$f(x) = x^3 + 4x^2 - 3x + 10, g(x) = x + 4$
- Find the remainder when $x^3+x^2-x+1$ is divided by $x+2$.
- Find the remainder using remainder theorem find the remainder when $x^3-6x^2+2x-4$ is divided by $1-\frac{3x}{2}$.
- Apply division algorithm to find the quotient $q(x)$ and remainder $r(x)$ on dividing $f(x)$ by $g(x)$ in the following: $f(x)\ =\ x^3\ –\ 6x^2\ +\ 11x\ –\ 6,\ g(x)\ =\ x^2\ +\ x\ +\ 1$
- Find the remainder when \( x^{3}-a x^{2}+6 x-a \) is divided by \( x-a \).
- Apply division algorithm to find the quotient $q(x)$ and remainder $r(x)$ on dividing $f(x)$ by $g(x)$ in the following:$f(x)\ =\ 15x^3\ –\ 20x^2\ +\ 13x\ –\ 12,\ g(x)\ =\ x^2\ –\ 2x\ +\ 2$
- On dividing $x^3 - 3x^2 + x + 2$ by a polynomial $g(x)$, the quotient and remainder were $x - 2$ and $-2x + 4$, respectively. Find $g(x)$.
- Apply division algorithm to find the quotient $q(x)$ and remainder $r(x)$ on dividing $f(x)$ by $g(x)$ in the following:$f(x)\ =\ 4x^3\ +\ 8x^2\ +\ 8x\ +\ 7,\ g(x)\ =\ 2x^2\ –\ x\ +\ 1$
- divide the polynomial $p( x)$ by the polynomial $g( x)$ and find the quotient and remainder in each of the following: $( p(x)=x^{3}-3 x^{2}+5 x-3$, $g(x)=x^{2}-2$.
- Divide the polynomial $p(x)$ by the polynomial $g(x)$ and find the quotient and remainder, in each of the following:(i) $p(x) = x^3 - 3x^2 + 5x -3, g(x) = x^2-2$(ii) $p(x) =x^4 - 3x^2 + 4x + 5, g(x) = x^2 + 1 -x$(iii) $p(x) = x^4 - 5x + 6, g(x) = 2 -x^2$
- Find the remainder when $x^3+ 3x^2 + 3x + 1$ is divided by \( x \)
- Find the remainder when $x^3+ 3x^2 + 3x + 1$ is divided by \( 5+2 x \)
Kickstart Your Career
Get certified by completing the course
Get Started
Data Structure
Networking
RDBMS
Operating System
Java
MS Excel
iOS
HTML
CSS
Android
Python
C Programming
C++
C#
MongoDB
MySQL
Javascript
PHP
Physics
Chemistry
Biology
Mathematics
English
Economics
Psychology
Social Studies
Fashion Studies
Legal Studies