Mathematics · Quantitative Aptitude

Algebraic Expressions and Polynomials

292 Questions

Algebraic expressions and polynomials form the foundation of algebra, involving variables, constants, and mathematical operations. This topic is heavily tested in the quantitative aptitude sections of SSC, banking, and state exams. Use these questions to practice expanding, factoring, and simplifying various polynomial expressions.

Expanding algebraic expressionsFactoring polynomialsAlgebraic identitiesFinding common factorsSimplifying numerical statements

Algebraic Expressions and Polynomials Questions

Multiple choice composition of ratios types of ratios ratio and proportions ratio and proportion maths

Find the compounded ratio of $(x^{2} - y^{2}) : (x^{2} + y^{2})$ and $(x^{4} - y^{4}) : (x + y)^{4}$

  1. $(x - y)^{3} : (x + y)^{3}$
  2. $(x + y)^{2} : (x^{2} - y^{2})$
  3. $1 : 1$
  4. $(x - y)^{2} : (x + y)^{2}$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

By the definition of compounded ratio these ratio can be expressed as
$\dfrac {(x^{2} - y^{2})}{(x^{2} + y^{2})} \times \dfrac {(x^{4} - y^{4})}{(x + y)^{4}}$
$\dfrac {(x^{2} - y^{2})}{(x^{2} + y^{2})} \times \dfrac {(x^{2} - y^{2})(x^{2} + y^{2})}{(x + y)^{4}}$
$= \dfrac {(x^{2} - y^{2})^{2}}{(x + y)^{4}}$
$= \dfrac {[(x - y) (x + y)]^{2}}{(x + y)^{4}}$
$= \dfrac {(x - y)^{2}}{(x + y)^{2}}$
Hence $(x - y)^{2} : (x + y)^{2}$

Multiple choice

3√x2

  1. x3/2

  2. x2/3

  3. x-2/3

  4. 3/x2
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The cube root of x^2 is written as x^(2/3) using fractional exponents.

Multiple choice

express y1/4 with a root sign

  1. √y

  2. ∛y

  3. ∜y

  4. √y4
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The fractional exponent 1/n corresponds to the nth root. Therefore, y^(1/4) is the fourth root of y, written as ∜y.

Multiple choice

5√x3y5

  1. 5√x3y5

  2. 5xy√x2y3

  3. 5xy2√xy

  4. 5x3y5
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

5 * sqrt(x^3 * y^5) = 5 * sqrt(x^2 * x * y^4 * y) = 5 * x * y^2 * sqrt(x * y).

Multiple choice

Express the answers with positive indices: (2m2n-4)/(6m5n-1)

  1. 1/(3m3n3)

  2. 3m3n3

  3. 6m7n5

  4. 2m7n5
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Divide the coefficients (2/6 = 1/3) and subtract the exponents for like bases (m^(2-5) = m^-3 and n^(-4 - -1) = n^-3). Since the indices must be positive, move the negative powers to the denominator to get 1/(3m^3n^3).

Multiple choice

2x3y24x½x4y4

  1. 16x4y6

  2. 4x4y6

  3. 4x8y6

  4. 8x8y6
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Multiply the coefficients (2 * 4 * 1/2 = 4) and add the exponents for like bases (x^(3+1+4) = x^8 and y^(2+4) = y^6). The result is 4x^8y^6.

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

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$

  1. Yes

  2. No

  3. Ambiguous

  4. Data insufficient
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Given f(y)=$3y^{4}+5y^{3}-7y^{2}+2y+2$ and (g)=$ y^{2}-4y+2$

Then $ y^{2}+3y+1\div 3y^{4}+5y^{3}-7y^{2}+2y+2\setminus 3y^{2}-y+7$
                                   $3y^{4}+9y^{3}+3y^{2}$
                                      -          -           -
                                  ---------------------------------------
                                       $-4y^{3}-10y^{2}+2y+2$
                                        $-4y^{3}-12y^{2}-4y+2$
                                            +          +           +
                                     -----------------------------------------
                                            $2y^{2}+6y+2$
                                             $2y^{2}+6y+2$
                                                -          -       -
                                    --------------------------------------------
                                                            0
So g(Y) is a factor of f(y).                     
                                             

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 $A=2x^{3}+5x^{2}+4x+1$ and $B=2x^{2}+3x+1$, then find the quotient from the following four option, when A is divided by B.

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


$\dfrac{A}{B} = \dfrac{2x^{3}+5x^{2}+4x+1}{2x^{2}+3x+1}$

$=\dfrac{2x^{3}+(3x^{2}+2x^{2})+(x+3x)+1}{2x^{2}+2x+1}$

$=\dfrac{(2x^{3}+3x^{2}+x)+(2x^{2}+3x+1)}{2x^{2}+3x+1}$

$=\dfrac{(2x^{2}+3x+1)(x+1)}{2x^{2}+3x+1}=x+1$

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

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

  1. 5 and -3

  2. 17 and -8

  3. 7 and-18

  4. 23 and -19
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Here, $x + 2$ is a factor of $x^3 + 10x^2 + mx + n$
$x =-2$
$(-2)^3 + 10(-2)^2 + m(-2) + n =0$
$ -8 + 40 = 2m - n $
$2m -n = 32$                     .....(i)
Again, $x-1 $ is a factor of $x^3 + 10x^2 + mx + n$
$x =1$
$1+10+m+n=0$
$m + n =-11$                    .....(ii)
Adding (i) and (ii). we get,
$3m = 21$
$m=7$
By putting m in (i). we get,
$2(7) - n = 32 $
$ - n = 18 $
   $n = -18$
Option C is correct.

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 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.

  1. $-3, 8$
  2. $3,-8$
  3. $-3,-8$
  4. $3, 8$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

f(x)=$ 2x^3+ax^2-bx+3$ 
At x=2 
f(2)=15 
f(1)=0 
f(x)=$ 2x^3+ax^2-bx+3$ 
f(1)= 2+a-b+3=0 
a-b+5=0......A 
f(x)= $2x^3+ax^2-bx+3$ 
f(2)=$ 2(2^3)+a(2^2)-2b+3=15 $
4a-2b=-4 
Multiply A by 2 and subtract from above equation 
4a-2b=-4 
2a-2b+10=0 
2a-10=-4 
2a= 6 
a=3 
From A 
3-b+5=0 
8-b=0 
b=8 
So a=3 and b=8

Multiple choice maths the plane equation of a plane in intercept form equation of a plane in different forms different forms of equation of a plane

The expression of $x+y+z=1$ in form of $x\cos { \alpha  } +y\cos { \beta  } +z\cos { \gamma  } =p$ is _______.

  1. $x+y+z=1$
  2. $\cfrac { x }{ 2\sqrt { 3 } } +\cfrac { y }{ 2\sqrt { 3 } } +\cfrac { z }{ 2\sqrt { 3 } } =\cfrac { 1 }{ \sqrt { 3 } } $
  3. $\cfrac { x }{ \sqrt { 3 } } +\cfrac { y }{ \sqrt { 3 } } +\cfrac { z }{ \sqrt { 3 } } =1$
  4. $\cfrac { x }{ \sqrt { 3 } } +\cfrac { y }{ \sqrt { 3 } } +\cfrac { z }{ \sqrt { 3 } } =\cfrac { 1 }{ \sqrt { 3 } } $
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

$\cfrac { x }{ \sqrt { 3 }  } +\cfrac { y }{ \sqrt { 3 }  } +\cfrac { z }{ \sqrt { 3 }  } =1$
$\quad \rightarrow P=\cfrac { \left| -1 \right|  }{ \sqrt { 1+1+1 }  } =\cfrac { 1 }{ \sqrt { 3 }  } $
$\therefore x+y+z=1$
$\therefore \cfrac { x }{ \sqrt { 3 }  } +\cfrac { y }{ \sqrt { 3 }  } +\cfrac { z }{ \sqrt { 3 }  } =\cfrac { 1 }{ \sqrt { 3 }  } $

Multiple choice maths power and exponent power of powers laws of exponents and powers law of indices

Which of the following represents the power of product rule?

  1. $(x\times y)^{a} = x^{a} \times y$
  2. $(x\times y)^{a} = x \times y^{a}$
  3. $(x\times y)^{a} = x^{a} + y^{a}$
  4. $(x\times y)^{a} = x^{a} \times y^{a}$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

If the product of the bases is powered  by the same exponent, 

then the result is multiplication of all the bases, 
each powered by the given exponent.
$\therefore (x\times y)^{a} = x^{a} \times y^{a}$.
So, option $D$ is correct.

Multiple choice maths power and exponent power of powers laws of exponents and powers law of indices

The rationalising factor of $\sqrt[5]{a^2b^3c^4}$ is _____.

  1. $\sqrt[5]{a^3b^2c}$
  2. $\sqrt[5]{a^3bc}$
  3. $\sqrt[5]{a^3b^2c^5}$
  4. $\sqrt[5]{a^3b^6c}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

To rationalize $(a^2b^3c^4)^{\frac{1}{5}}$, fifth root must be removed, 

$\therefore$We should multiply it by the factor $(a^3b^2c)^{\frac{1}{5}}$, So thst it will become $abc$.