Linear and synthetic method of division - class-IX
Description: linear and synthetic method of division | |
Number of Questions: 64 | |
Created by: Sundari Chatterjee | |
Tags: remainder and factor theorems polynomials and factorization multiplication and division of algebraic expressions maths polynomials basic algebra |
If $P(x)$ and $Q(x)$ are two polynomial such that $f(x)=P(x^3)+Q(x^3)$ is divisible by $x^2+x+1$, then?
The remainder when $3x^{4}+5x^{2}+9$ is divided by $x^{2}-2$ is
What is the degree of the remainder atmost, when a fourth degree polynomial is divided by a quadratic polynomial?
Can $(x - 1)$ be the remainder on division of a polynomial $p(x)$ by $2x + 3$?
If quotient = $3x^2\, -\, 2x\, +\, 1$, remainder = $2x - 5$ and divisor = $x + 2$, then the dividend is:
The remainder if $a{x}^{3}+b{x}^{2}+cx+d$ is divided by $ax+b$
If $ \displaystyle 2x^{3}+4x^{2}+2ax+b $ is exactly divisible by $ \displaystyle x^{2}-1 $ Then the value of $a$ and $b$ respectively will be
The product of $x^2y$ and $\cfrac{x}{y}$ is equal to the quotient obtained when $x^2$ is divided by ____.
There is a remainder of 3 when a number is divided by 6. What will be the remainder if the square of the same number is divided by 6?
If on dividing a non-zero polynomial $p(x)$ by a polynomial $g (x)$, the remainder is zero, what is the relation between the degrees of $p(x)$ and $g (x)$?
If the polynomial $x^3-x^2+x-1$ is divided by $x-1$, then the quotient is :
Find the reminder when ${x^3} + 3{x^2} + 3x + 1$ is divided by $x + \pi $
When the polynomial ${x^4} + {x^2} + 1$ is divided by $(x + 1)({x^2} - x + 1)$ then the remainder is $ax + b$ , then $a + b$ is equal to
Find the quotient $q(x)$ and remainder $r(x)$ of the following when $f(x)$ is divided by $g(x)$.
$p(x)=x^3-3x^2-x+3$;
$g(x)=x^2-4x+3$
Find the quotient $q(x)$ and remainder $r(x)$ of the following when $f(x)$ is divided by $g(x)$.
$p(x)=x^6+x^4-x^2-1$;
$g(x)=x^3-x^2+x-1$
Check whether $g(y)$ is a factor of $f(y)$ by applying the division algorithm.
$f(y)=3y^4+5y^3-7y^2+2y+2$
$ g(y)=y^2+3y+1$
Divide the polynomial $p(x)$ by the polynomial $g(x)$ and find the quotient and remainder.
$p(x)=x^4-3x^2+4x+5$
$g(x)=x^2+1-x$
Polynomials $p(x), g(x), q(x)$ and $r(x)$, which satisfy the division algorithm and deg $r(x)=0$, are
On dividing $f(x)$ by a polynomial $x-1-x^2$, the quotient $q(x)$ and remainder $r(x)$ are $(x-2)$ and $3$ respectively. Then $f(x)$ is
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 $g(x)$.
Polynomials $p(x), g(x), q(x)$ and $r(x)$, which satisfy the division algorithm and "deg $q(x) = $ deg $ r(x)$", are
On dividing $f(x)=2x^5+3x^4+4x^3+4x^2+3x+2$ by a polynomial $g(x)$, where $g(x)=x^3+x^2+x+1$, the quotient obtained as $2x^2+x+1$. Find the remainder $r(x)$.
Polynomials $p(x), g(x), q(x)$ and $r(x)$, which satisfy the division algorithm and "deg $p(x) = $ deg $q(x)$" are
What should be added to $8x^4+14x^3-2x^2+7x-8$ so that the resulting polynomial is exactly divisible by $4x^2+3x-2$?
Check whether the first polynomial is a factor of the second polynomial by applying the division algorithm. $x^3-3x+1, x^5-4x^3+x^2+3x+1$
If the polynomial $f(x)=x^4-6x^3+16x^2-25x+10$ is divided by another polynomial $x^2-2x+k$, the remainder comes out to be $(x+a)$, then values of $k$ and $a$ are
Find the value of $b$ for which the polynomial $2x^3+9x^2-x-b$ is exactly divisible by $2x+3$?
What must be subtracted from or added to $8x^4+14x^3-2x^2+8x-12$ so that it may be exactly divisible by $4x^2+3x-2$?
$\displaystyle \left( { 3x }^{ 2 }-x \right) \div \left( -x \right) $ is equal to
A polynomial when divided by $\displaystyle \left ( x-6 \right )$ gives a quotient $\displaystyle x^{2}+2x-13$ and leaves a remainder $-8$. Then polynomial is
The expression that should be subtracted from $\displaystyle 4x^{4}-2x^{3}-6x^{2}+x-5$ so that is may be exactly divisible by $\displaystyle 2x^{2}+x-2$ is
For a polynomial, dividend is $\displaystyle x^{4}+4x-2x^{2}+x^{3}-10$, quotient is $\displaystyle x^{2}+3x-3x^{2}+4x+12$ and remainder is $14$, then divisor is equal to
If $\displaystyle \left ( x^{2}+4x-21 \right )$ is divided by $x + 7$ then the quotient is
If $\displaystyle f(x)=x^{4}-2x^{3}+3x^{2}-ax+b$ is a polynomial such that when it is divided by $( x - 1 )$ and $( x +1)$, the remainders are $5$ and $19 $ respectively, the remainder when $f(x)$ is divisible by $(x -2 ) $ is
When a number is divided by $13$, the remainder is $11$. When the same number is divided by $17$, the remainder is $9$. What is the number ?
On dividing a number by $56$, we get $29$ as remainder. On dividing the same number by $8$, what will be the remainder ?
The remainders of polynomial f(x) when divided by x-1, x-2 are 2,3 then the remainder of f(x) when divided by (x-1) (x-2) is
If the remainders of the polynomial f(x) when divided by x+1 and x-1 are 3, 7 then the remainder of f(x) when divided by $(x^{2} -1 )$ is
Given $f(x)$ is a cubic polynomial in $x$. If $f(x)$ is divided by $(x + 3), (x + 4), (x + 5)$ and $(x + 6)$ then it leaves the remainders $0, 0, 4$ and $6$ respectively. Find the remainder when $f(x)$ is divided by $x + 7$.
Find the remainder when $-2x^3-2x^2+27x-30$ is divided by $2-x$.
$p(x)=(x^2-10x-24)$ , when divided by $x+2$ and $x\neq -2$ gives the quotient $Q$. Find $Q$.
When $(x^3-2x^2+px-q)$ is divided by $(x^2-2x-3)$, the remainder is $(x-6)$. The values of $p$ and $q$ respectively are ____.
When a polynomial $P(x)$ is divided by $x, (x - 2)$ and $(x - 3)$, remainders are $1$, $3$ and $2$ respectively. the same polynomial is divided by $x(x - 2)(x - 3)$, the remainder is $ax^2 + bx + c$, then the value of $c$ is
The quotient and remainder when $x^{2002}$ $- 2001$ is divided by $x^{91}$ are
Which of the following given options is/ are correct?
If $p(x)=q(x)g(x)+r(x)$ (By Division Algorithm) where p(x), g(x) are any two polynomials with $g(x)\neq 0$, then
A polynomial $f(x)$ with rational coefficient leaves reminder $15$, when divided by $(x-3)$ and remainder $2x+1$, when divided by $(x-1)^{2}$. If $p$ is coefficient of $x$ of its remainder which will come out if $f(x)$ is divided by $(x-3)(x-1)^{2}$ then find $p$.
The remainder obtained when the polynomial $1+x+x^ {3}+x^ {9}+x^ {27}+x^ {81}+x^ {243}$ is divisible by $x-1$ is
If $A=2x^{3}+5x^{2}+4x+1$ and $B=2x^{2}+3x+1$, then find the quotient from the following four option, when A is divided by B.
Evaluate: $\displaystyle \frac{a^3\, +\, b^3\, +\, c^3\, -\, 3abc}{a^2\, +\, b^2\, +\, c^2\, -\, ab\, -\, bc\, -\, ca}$
If x + 2 and x-1 are the factors of $x^3 + 10x^2+mx + n$, then the values of m and n are respectively
The expression $2x^3 + ax^2 + bx +3$, where a and b are constants, has a factor of x-1 and leaves a remainder of 15 when divided by x+2. Find the value of a and b respectively.
If $(x^{3} + 5x^{2} + 10k)$ leaves remainder $-2x$ when divided by $(x^{2} + 2)$, then what is the value of k?
The polynomial $f(x)={ x }^{ 4 }-2{ x }^{ 3 }+3{ x }^{ 2 }-ax+b$ when divided by $(x-1)$ and $(x+1)$ leaves the remainders $5$ and $19$ respectively. Find the values of $a$ and $b$. Hence, find the remainder when $f(x)$ is divided by $(x-2)$
$32x^{10}-33x^{5}+1$ is divisible by
The remainder, when $({ 15 }^{ 23 }+{ 23 }^{ 23 })$ is divided by $19$, is
If $(x^{100} + 2x^{99} + K)$ is exactly divisible by $(x + 1)$, find the value of 'K'
The sum of the digits of a 3 digit number is subtracted from the number. The resulting number is always.
The remainder when the polynomial $1+x^2+x^4+x^6+....+x^{22}$ is divided by $1+x+x^2+x^3+....+x^{11}$ is?
If the polynomial $x^{19}+x^{17}+x^{13}+x^{11}+x^7+x^5+x^3$ is divided by $(x^2+1)$, then the remainder is:
The decimal representation of $2005!$ ends with $m$ zeroes then $m=$
What will be the Remainder when $3x^{3} - 2x^{2} - 7x + 6$ is divided by $x + 1$?
A body falling from rest under gravity passes a certain point $P$.It was a distance of $400m$ from P and $4$ sec prior to passage through $P$ If $g=10m/sec^2$,then the height above the point $"P"$ from where the body began to fall is ?
The remainder when $x^3 + 4x^2 - 7x + 6$ is divided by $(x - 1)$ is
What will be the Quotient when $4x^{3} - 8x^{2} - x + 5$ is divided by $2x - 1$?