Give examples of polynomials $p(x), g(x), q(x)$ and $r(x)$, which satisfy the division algorithm and deg $r(x) = 0$
Given:
deg $r(x) = 0$
To do:
We have to give examples of polynomials $p(x), g(x), q(x)$ and $r(x)$, which satisfy the division algorithm and deg $r(x) = 0$
Solution:
$p(x), g(x), q(x), r(x)$
deg $r(x) = 0$
This is possible when product of $q(x)$ and $g(x)$ form a polynomial whose degree is equal to degree of $p(x)$ and constant term.
Related Articles
- Give examples of polynomials $p(x), g(x), q(x)$ and $r(x)$, which satisfy the division algorithm and deg $q(x) =$ deg $r(x)$
- Give examples of polynomials $p(x), g(x), q(x)$ and $r(x)$, which satisfy the division algorithm and(i) deg $p(x) =$ deg $q(x)$(ii) deg $q(x) =$ deg $r(x)$(iii) deg $r(x) = 0$
- Apply division algorithm to find the quotient $q(x)$ and remainder $r(x)$ on dividing $f(x)$ by $g(x)$ in the following: $f(x)\ =\ x^3\ –\ 6x^2\ +\ 11x\ –\ 6,\ g(x)\ =\ x^2\ +\ x\ +\ 1$
- Answer the following and justify:What will the quotient and remainder be on division of \( a x^{2}+b x+c \) by \( p x^{3}+q x^{2}+r x+s, p ≠ 0 ? \)
- Apply division algorithm to find the quotient $q(x)$ and remainder $r(x)$ on dividing $f(x)$ by $g(x)$ in the following:$f(x)\ =\ 4x^3\ +\ 8x^2\ +\ 8x\ +\ 7,\ g(x)\ =\ 2x^2\ –\ x\ +\ 1$
- Apply division algorithm to find the quotient $q(x)$ and remainder $r(x)$ on dividing $f(x)$ by $g(x)$ in the following:$f(x)\ =\ 15x^3\ –\ 20x^2\ +\ 13x\ –\ 12,\ g(x)\ =\ x^2\ –\ 2x\ +\ 2$
- Apply division algorithm to find the quotient $q(x)$ and remainder $r(x)$ on dividing $f(x)$ by $g(x)$ in the following:$f(x)\ =\ 10x^4\ +\ 17x^3\ –\ 62x^2\ +\ 30x\ –\ 3,\ g(x)\ =\ 2x^2\ +\ 7x\ +\ 1$
- In \( \Delta X Y Z, X Y=X Z \). A straight line cuts \( X Z \) at \( P, Y Z \) at \( Q \) and \( X Y \) produced at \( R \). If \( Y Q=Y R \) and \( Q P=Q Z \), find the angles of \( \Delta X Y Z \).
- 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) $p(x) = x^4 - 5x + 6, g(x) = 2 -x^2$
- Use the Factor Theorem to determine whether \( g(x) \) is a factor of \( p(x) \) in each of the following cases:(i) \( p(x)=2 x^{3}+x^{2}-2 x-1, g(x)=x+1 \)(ii) \( p(x)=x^{3}+3 x^{2}+3 x+1, g(x)=x+2 \)(iii) \( p(x)=x^{3}-4 x^{2}+x+6, g(x)=x-3 \)
- Identify constant, linear, quadratic and cubic polynomials from the following polynomials:\( r(x)=3 x^{3}+4 x^{2}+5 x-7 \)
- divide the polynomial $p( x)$ by the polynomial $g( x)$ and find the quotient and remainder in each of the following: $( p(x)=x^{3}-3 x^{2}+5 x-3$, $g(x)=x^{2}-2$.
- For which values of \( a \) and \( b \), are the zeroes of \( q(x)=x^{3}+2 x^{2}+a \) also the zeroes of the polynomial \( p(x)=x^{5}-x^{4}-4 x^{3}+3 x^{2}+3 x+b \) ? Which zeroes of \( p(x) \) are not the zeroes of \( q(x) \) ?
- Divide the polynomial $p(x)$ by the polynomial $g(x)$ and find the quotient and remainder, in each of the following:$p(x) = x^4 - 5x + 6, g(x) = 2 -x^2$
- Divide the polynomial $p(x)$ by the polynomial $g(x)$ and find the quotient and remainder, in each of the following:$p(x) =x^4 - 3x^2 + 4x + 5, g(x) = x^2 + 1 -x$
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