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 ? $
To do:
We have to find the quotient and remainder on division of \( a x^{2}+b x+c \) by \( p x^{3}+q x^{2}+r x+s, p
≠ 0 \).
Solution:
Here,
Divisor $=px3 + qx2 + rx + s, p≠0$
Dividend $= ax^2 + bx + c$
We observe that,
Degree of divisor $>$ Degree of dividend
We know that,
If the degree of dividend is less than the degree of the divisor, then the quotient will be zero and the remainder is same as the dividend.
Therefore, by division algorithm,
Quotient $= 0$ and Remainder $= ax^2 + bx + c$
Related Articles
- 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) $p(x) = x^4 - 5x + 6, g(x) = 2 -x^2$
- Apply division algorithm to find the quotient $q(x)$ and remainder $r(x)$ on dividing $f(x)$ by $g(x)$ in the following: $f(x)\ =\ x^3\ –\ 6x^2\ +\ 11x\ –\ 6,\ g(x)\ =\ x^2\ +\ x\ +\ 1$
- Divide the polynomial $p(x)$ by the polynomial $g(x)$ and find the quotient and remainder, in each of the following:$p(x) = x^4 - 5x + 6, g(x) = 2 -x^2$
- Apply division algorithm to find the quotient $q(x)$ and remainder $r(x)$ on dividing $f(x)$ by $g(x)$ in the following:$f(x)\ =\ 15x^3\ –\ 20x^2\ +\ 13x\ –\ 12,\ g(x)\ =\ x^2\ –\ 2x\ +\ 2$
- Answer the following and justify:Can \( x^{2}-1 \) be the quotient on division of \( x^{6}+2 x^{3}+x-1 \) by a polynomial in \( x \) of degree 5 ?
- Divide the polynomial $p(x)$ by the polynomial $g(x)$ and find the quotient and remainder, in each of the following:$p(x) =x^4 - 3x^2 + 4x + 5, g(x) = x^2 + 1 -x$
- Find the zero of the polynomial in each of the following cases:(i) \( p(x)=x+5 \)(ii) \( p(x)=x-5 \)(iii) \( p(x)=2 x+5 \)(iv) \( p(x)=3 x-2 \)(v) \( p(x)=3 x \)(vi) \( p(x)=a x, a ≠0 \)(vii) \( p(x)=c x+d, c ≠0, c, d \) are real numbers.
- Apply division algorithm to find the quotient $q(x)$ and remainder $r(x)$ on dividing $f(x)$ by $g(x)$ in the following:$f(x)\ =\ 4x^3\ +\ 8x^2\ +\ 8x\ +\ 7,\ g(x)\ =\ 2x^2\ –\ x\ +\ 1$
- 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:$f(x)\ =\ 10x^4\ +\ 17x^3\ –\ 62x^2\ +\ 30x\ –\ 3,\ g(x)\ =\ 2x^2\ +\ 7x\ +\ 1$
- Give examples of polynomials $p(x), g(x), q(x)$ and $r(x)$, which satisfy the division algorithm and deg $r(x) = 0$
- 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 \( q(x) \) ?
- Verify whether the following are zeroes of the polynomial, indicated against them.(i) \( p(x)=3 x+1, x=-\frac{1}{3} \)(ii) \( p(x)=5 x-\pi, x=\frac{4}{5} \)(iii) \( p(x)=x^{2}-1, x=1,-1 \)(iv) \( p(x)=(x+1)(x-2), x=-1,2 \)(v) \( p(x)=x^{2}, x=0 \)(vi) \( p(x)=l x+m, x=-\frac{m}{l} \)(vii) \( p(x)=3 x^{2}-1, x=-\frac{1}{\sqrt{3}}, \frac{2}{\sqrt{3}} \)(viii) \( p(x)=2 x+1, x=\frac{1}{2} \)
- 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) \( p(x)=x^{3}-4 x^{2}+x+6, g(x)=x-3 \)
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