Quantitative Aptitude · Mathematics

Number and Polynomial Division

133 Questions

Number and polynomial division tests your skill in finding quotients, remainders, and applying theorems to algebraic expressions. These calculations are a staple in quantitative aptitude exams. Solving these problems enhances speed and accuracy for competitive tests.

Number remaindersPolynomial divisionQuotient and remainderFactor and remainder theoremSuccessive division

Number and Polynomial Division Questions

Multiple choice maths remainder and factor theorems linear and synthetic method of division factorising expressions using factor theorem division algorithm for polynomials

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 ?

  1. $853$
  2. $278$
  3. $349$
  4. $670$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

$x = 13p + 11$ and $x = 17q + 9$
$\therefore$ $13p + 11 = 17q + 9$
$\therefore$ $17q - 13p = 2$
$\therefore$ q $=\dfrac{2 + 13p}{17}$
The least value of p for which q $=\dfrac{2 + 13p}{17}$ is a whole number is $p = 26$
x $= (13 \times 26 + 11)$
$= (338 + 11)$
$= 349$

Multiple choice maths remainder and factor theorems linear and synthetic method of division factorising expressions using factor theorem division algorithm for polynomials

On dividing a number by $56$, we get $29$ as remainder. On dividing the same number by $8$, what will be the remainder ?

  1. $4$
  2. $5$
  3. $6$
  4. $7$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Applying remainder theorem i.e. $A=bq+r $
where, $b$ = divisior 
$r$ = remainder 
$\therefore A = 56q + 29$
if q = 1 
The no. is $A = 85$
On dividing by $8$, Remainder $(r) =5$

Multiple choice maths remainder and factor theorems linear and synthetic method of division factorising expressions using factor theorem division algorithm for polynomials

Find the remainder when  $-2x^3-2x^2+27x-30$ is divided by $2-x$.

  1. $0$
  2. $1$
  3. $2$
  4. $3$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation
Quotient----->  $2x^2+6x-15$
 -x + 2   $-2x^2-2x^2+27x-30$  $-2x^3+4x^2$
    $-6x^2+27x-30$ $-6x^2+12x$-----------------------------------$15x-30$$15x-30$-------------------0
 
 
Multiple choice maths remainder and factor theorems linear and synthetic method of division factorising expressions using factor theorem division algorithm for polynomials

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 ____. 

  1. $-2, 7$
  2. $2, -6$
  3. $-2, 6$
  4. $2, 6$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

$x^2-2x-3=(x+1)(x-3)$

$x^3-2x^2+px-q-(x-6)=0$ at $x=-1$ and $x=3$.         ($\because (x-6)$ is the remainder)
put $x=-1$,
$-1-2-p-q+1+6=0\ \Rightarrow p+q=4\dots eqn (1)$
Now, put $x=3$
$27-18+3p-q-3+6=0\ \Rightarrow 3p-q=-12\dots eqn (2)$
Add equation 1 and 2, we get
$4p=-8\Rightarrow p=-2$
 and $q=4-p=6$

Multiple choice maths remainder and factor theorems linear and synthetic method of division factorising expressions using factor theorem division algorithm for polynomials

The quotient and remainder when $x^{2002}$ $- 2001$ is divided by $x^{91}$ are 

  1. $x^{91 \times 22}, 2001$
  2. $x^{91}, 2001$
  3. $x^{91\times 21}, -2001$
  4. $x^9, -2001$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

$2002 = 91 \times 22$

$\therefore x^{2002} = x^{91 \times 22}$
$\therefore$ $x^{2002} - 2001$ $=$ $x^{91} \times (x^{91 \times 21}) - 2001$

When $x^{91} \times (x^{91 \times 21}) - 2001$ is divided by $x^{91}$, then
Quotient $= x^{91 \times 21}$ 
And 
Remainder $= -2001$

Multiple choice maths multiplication and division of algebraic expressions linear and synthetic method of division factorising expressions using factor theorem division algorithm for polynomials

The remainder obtained when the polynomial $1+x+x^ {3}+x^ {9}+x^ {27}+x^ {81}+x^ {243}$ is divisible by $x-1$ is

  1. $3$
  2. $5$
  3. $7$
  4. $11$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

$p{\left( x \right)} = 1 + x + {x}^{3} + {x}^{9} + {x}^{27} + {x}^{81} + {x}^{243}$

Let $q{\left( x \right)}$ be the quotient when $P{\left( x \right)}$ divided by $\left( x - 1 \right)$.
Therefore,
$P{\left( x \right)} = \left( x - 1 \right) \cdot q{\left( x \right)} + A$
$P{\left( 1 \right)} = \left( 1 - 1 \right) \cdot q {\left( x \right)} + A$
$7 = 0 + A$
$A = 7$
Hence the remainder is $7$.

Multiple choice maths multiplication and division of algebraic expressions linear and synthetic method of division factorising expressions using factor theorem division algorithm for polynomials

If $(x^{3} + 5x^{2} + 10k)$ leaves remainder $-2x$ when divided by $(x^{2} + 2)$, then what is the value of k?

  1. $-2$
  2. $-1$
  3. $1$
  4. $2$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

$x^{3} + 5x^{2} + 10k$
$= (x^{2} + 2)(x + 5) + 10k - 2x - 10$
$\Rightarrow 10k - 2x - 10 = -2x$
$\Rightarrow 10k - 10 = 0$ or $k = 1$.

Multiple choice maths multiplication and division of algebraic expressions linear and synthetic method of division factorising expressions using factor theorem division algorithm for polynomials

The remainder, when $({ 15 }^{ 23 }+{ 23 }^{ 23 })$ is divided by $19$, is 

  1. $4$
  2. $17$
  3. $23$
  4. $0$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation
Given, $(15)^{23}+(23)^{23}$

$=(19-4)^{23}+(19+4)^{23}$

$\Rightarrow $ In bino  expansion  of above expression the term containing  19 well be cancelled 
as they disable by 19 then remaining term are 

$\Rightarrow  (-4)^{23}+(4)^{23}=0$

Therefore the remainder is exactly  zero .
Multiple choice maths multiplication and division of algebraic expressions linear and synthetic method of division factorising expressions using factor theorem division algorithm for polynomials

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?

  1. $0$
  2. $2$
  3. $1+x^2+x^4+...+x^{10}$
  4. $2(1+x^2+x^4+....+x^{10})$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation
$ \left( \sum _{n=0}^N x^{n} \right) = \left ( \dfrac{x^{N+1}-1}{x-1} \right ) $

$ \Rightarrow Dividend = \left ( \dfrac{x^{24}-1}{x^{2}-1} \right ) $
$ Divisor = \left ( \dfrac{x^{12}-1}{x-1} \right ) $

Now,
$ \left ( \dfrac{x^{24}-1}{x^{2}-1} \right )  = \left ( \dfrac{x^{12}-1}{x-1} \right ) \left ( \dfrac{x^{12}-1+2}{x+1} \right ) = \left ( \dfrac{\left ( x^{12}-1 \right )^{2}}{x^{2}-1} \right ) + 2\left ( \dfrac{x^{12}-1}{x^{2}-1} \right ) $

$ \Rightarrow Remainder = 2\left ( 1+x^{2}+x^{4}...+x^{10} \right ) $
Multiple choice maths multiplication and division of algebraic expressions linear and synthetic method of division factorising expressions using factor theorem division algorithm for polynomials

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:

  1. $1$
  2. $x^2+4$
  3. $-x$
  4. $x$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Divide the polynomial by x^2+1 by substituting x^2 = -1. The terms become powers of (-1), simplifying to -x.

Multiple choice maths polynomials linear and synthetic method of division factorising expressions using factor theorem division algorithm for polynomials

The remainder when $x^3 + 4x^2 - 7x + 6$ is divided by $(x - 1)$ is

  1. $4$
  2. $0$
  3. $-4$
  4. $3$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Let $f\left( x \right) =x^{ 3 }+4x^{ 2 }-7x+6$
As $f\left( x \right) $ is divided by $x-1$, substituting $x=1$ in $f\left( x \right) $ we get
$f\left( 1 \right) =1^{ 3 }+4\cdot1^{ 2 }-7\cdot1+6=4$
Hence, $4$ is the remainder.

Multiple choice maths polynomials linear and synthetic method of division factorising expressions using factor theorem division algorithm for polynomials

What will be the Quotient when $4x^{3} - 8x^{2} - x + 5$ is divided by $2x - 1$?

  1. $2x^{2} - 3x - 2$
  2. $3x^{2} - 6x - 2$
  3. $4x^{2} - 6x +4$
  4. $2x^{2} - 6x - 2$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Given: equation $4x^3-8x^2-x+5$

To find the quotient when divided by $2x-1$
Sol: $2x-1)\overline{4x^3-8x^2-x+5}(2x^2-3x-2)\\quad\quad \quad 4x^3-2x^2\\quad\quad\quad \overline{\quad \quad -6x^2-x}\\quad\quad\quad\quad\quad- 6x^2+3x\\quad\quad\quad\overline{\quad\quad\quad\quad \quad -4x+5}\\quad\quad\quad\quad\quad\quad \quad \quad- 4x+2\\quad\quad\quad\quad\overline {\quad\quad\quad\quad\quad\quad\quad 3} $

Multiple choice multiplication and division vedic methods of multiplication vedic mathematics history of mathematics maths

Perform division of $422\div 11$ using Nikhilam Sutra Method on base $10$. Also, find the quotient$(Q)$ and remainder$(R)$.

  1. $Q=40$ and $R=3$
  2. $Q=39$ and $R=5$
  3. $Q=33$ and $R=3$
  4. $Q=38$ and $R=4$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

For 422/11, base 10, divisor 11 has a deviation of 1. Using Nikhilam division, 422 / 11 = 38 with remainder 4.