Mathematics · Quantitative Aptitude

Algebraic Expressions and Polynomials

275 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.