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 maths be my multiple, i'll be your factor co-prime numbers lcm lowest common multiple (l.c.m.)

Find the common factors of the given terms:

$16 x^3, 4x^2, 32x$

  1. $x$
  2. $16$
  3. $4x$
  4. $32$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

$16 x^3$,  $4x^2$, $32x$
The factors of $16 x^3=2\times 2\times 2\times 2\times x\times x\times x$
The factors of $4x^2=2\times 2 \times x\times x$
The factors of $32x=2\times 2\times 2\times 2\times 2\times x$
The common factors are $2\times 2 \times x=4x$

Multiple choice business economics and quantitative methods linear correlation spearman's coefficient of correlation spearman's rank correlation method correlation coefficients

What is correction factor(C.F) in the rank correlation coefficient.

  1. C.F $=\sum (m^{2}-1)$
  2. C.F $=\sum (m^{2}+1)$
  3. C.F $=\sum m^{2}(m^{2}-1)$
  4. C.F $=\sum m(m^{2}-1)$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

The correction factor in the rank correlation coefficient is given by $\sum m(m^2-1)$


where $m$ is the number of times the data repeats.
Hence, C.F $=\sum m(m^2-1)$.

Multiple choice maths vectors:planes in three dimensions cartesian equation of plane general form of the equation of a plane lines in space

The normal form of $2x-2y+z=5$ is

  1. $12x-4y+3z=39$
  2. <p class="MsoNormal">$\displaystyle \dfrac{-6}{7}x+\dfrac{2}{7}y+\dfrac{3}{7}z=1$</p>
  3. <p class="MsoNormal">$\displaystyle \dfrac{12}{13}x-\dfrac{-4}{13}y+\dfrac{3}{13}z=3$</p>
  4. <p class="MsoNormal">$\displaystyle \dfrac{2}{3}x-\dfrac{2}{3}y+\dfrac{1}{3}z=\dfrac{5}{3}$</p>
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

The dr's of the normal to the plane are $(2,-2,1)$.
The dc's will be $\left ( \dfrac{2}{3} , \dfrac{-2}{3} , \dfrac{1}{3} \right)$
Hence, the equation of the plane in the normal form will be,
$ \dfrac{2x}{3} - \dfrac{2y}{3} + \dfrac{z}{3} $ = $ \dfrac{5}{3} $

Multiple choice maths brackets order operations and algebra using brackets in algebraic expressions order of operations

Use the BODMAS rule to reduce the expression: $x-1[(x^2+x-2)(x^2-1^2)\div (x-1)^2]$.

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

$x-1[(x^2+x-2)(x^2-1^2)\div (x-1)^2]$
We need to follow BODMAS rule.
=> Brackets (parts of a calculation inside brackets always come first).
=> Orders (numbers involving powers or square roots).
=> Division.
=> Multiplication.
=> Addition.
=> Subtraction.
$=$ $\frac{x-1(x^2+x-2)(x+1)(x-1)}{(x-1(x-1)}$
$=$ $x^3+x^2-2x+x^2+x-2$
$=$ $x^3+2x^2-x-2$

Multiple choice maths brackets order operations and algebra using brackets in algebraic expressions order of operations

Expand the expression using BODMAS rule: $x^2-x[(-x)(-2+x)]\div x+x^3-3x^2$

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

$x^2-x[(-x)(-2+x)]\div x+x^3-3x^2$
We need to follow BODMAS rule.
=> Brackets (parts of a calculation inside brackets always come first).
=> Orders (numbers involving powers or square roots).
=> Division.
=> Multiplication.
=> Addition.
=> Subtraction.
$=$ $-2x^2-x[\dfrac{(-x)(-2+x)}{x}]+x^3$
$=$ $-2x^2-x[2-x]+x^3$
$=$ $x^3-2x^2-2x+x^2$
$=$ $x^3-x^2-2x$

Multiple choice maths brackets order operations and algebra using brackets in algebraic expressions order of operations

Reduce the following expression using BODMAS rule: $2y-1(y-y^2)+5y[(-2y)(y^2-1)]$

  1. $10y^4+11y^2+y$
  2. $-10y^4+11y^2-y$
  3. $-10y^4+11y^2+y$
  4. $-10y^4-11y^2+y$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

$2y-1(y-y^2)+5y[(-2y)(y^2-1)]$
We need to follow BODMAS rule.
=> Brackets (parts of a calculation inside brackets always come first).
=> Orders (numbers involving powers or square roots).
=> Division.
=> Multiplication.
=> Addition.
=> Subtraction.
$=$ $2y-y+y^2+5y[-2y^3+2y]$
$=$ $y+y^2-10y^4+10y^2$
$=$ $-10y^4+11y^2+y$

Multiple choice maths brackets order operations and algebra using brackets in algebraic expressions order of operations

$24[x+1]-[x^2-24+x]-[2x^2]\div [x^2]$ using BODMAS rule to reduce the expression.

  1. $x^2+23x+46$
  2. $-x^2+23x+46$
  3. $-x^2-23x+46$
  4. $-x^2+23x-46$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

$24[x+1]-[x^2-24+x]-[2x^2]\div [x^2]$
We need to follow BODMAS rule.
=> Brackets (parts of a calculation inside brackets always come first).
=> Orders (numbers involving powers or square roots).
=> Division.
=> Multiplication.
=> Addition.
=> Subtraction.
$=$ $24[x+1]-[x^2-24+x]-[2x^2]\div [x^2]$
$=$ $24x+24-x^2+24-x-2$
$=$ $-x^2+46+23x$
$=$ $-x^2+23x+46$

Multiple choice maths brackets order operations and algebra using brackets in algebraic expressions order of operations

Solve the expression using BODMAS rule: $3x(x-2)+x(x^2\times 2x)-12x$

  1. $2x^5-3x^2-18x$
  2. $2x^5+3x^2-18x$
  3. $2x^5+3x^2+18x$
  4. $-2x^5-3x^2-18x$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

$3x(x-2)+x(x^2\times 2x)-12x$
We need to follow BODMAS rule.
=> Brackets (parts of a calculation inside brackets always come first).
=> Orders (numbers involving powers or square roots).
=> Division.
=> Multiplication.
=> Addition.
=> Subtraction.
$=$ $3x(x-2)+x(x^2\times 2x)-12x$
$=$ $3x^2-6x+x^3\times2x^2-12x$
$=$ $3x^2-6x+2x^5-12x$
$=$ $2x^5+3x^2-18x$

Multiple choice properties of proportion ratio and proportions ratio and proportion maths

After applying invertendo to $3:7::2:9$ we get

  1. $3:7::2:9$
  2. $3:2::7:9$
  3. $9:7::2:3$
  4. $7:3::9:2$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

If $a : b :: c : d$ then $b : a :: d : c$ is invertendo property of ratios

We have $3:7::2:9$
After applying invertendo we get
$7:3::9:2$
Option D is correct

Multiple choice properties of proportion ratio and proportions ratio and proportion maths

Find the value of $a$ and $b$ respectively
After applying invertendo to $2:5::3:9$  we get $a:2::9:b$ 

  1. $9$ and $2$
  2. $2$ and $9$
  3. $3$ and $5$
  4. $5$ and $3$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

If $a : b :: c : d$ then $b : a :: d : c$ is invertendo property of ratios

We have $2:5::3:9$
After applying invertendo we get
$5:2::9:3\equiv a:2::9:b$
$\Rightarrow a=5,b=3$
Option D is correct

Multiple choice properties of proportion ratio and proportions ratio and proportion maths

After applying invertendo to $1:2::3:4$ we get

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

If $a : b :: c : d$ then $b : a :: d : c$ is invertendo property of ratios

We have $1:2::3:4$
After applying invertendo we get
$2:1::4:3$
Option B is correct

Multiple choice properties of proportion ratio and proportions ratio and proportion maths

After applying invertendo to $1:2::8:9$ we get:

  1. $2:1::9:8$
  2. $1:8::2:9$
  3. $8:9::1:2$
  4. $1:9::8:2$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

If $a : b :: c : d$ then $b : a :: d : c$ is invertendo property of ratios

We have $1:2::8:9$
After applying invertendo we get
$2:1::9:8$
Option A is correct

Multiple choice the nth roots of unity complex numbers maths

$(1-\omega +\omega^{2})(1-\omega^{2}+\omega^{4})(1-\omega^{4}+\omega^{8})......$to 2n factors =

  1. 2
  2. $2^{2n}$

  3. 2n

  4. none of these
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation
Given,

$(1−ω+ω^2)(1−ω^2+ω^4)(1−ω^4+ω^8)$...... to $2n$

we have,

$1+w+w^2=0$

and $w^3=1$

$\Rightarrow (-w-w)(-w^2-w^2)(-w-w)....................2n$

$=(-2w)(-2w^2)(-2w).............2n$

here, we can see, 2 consecutive terms are same and are even, so we get,

$=(4w^3)(4w^3)(4w^3).........2n$

$=4 \times 4 \times .........2n$

$=2^2.........2n$

for $2n$ terms, we get,

$=2^{2n}$