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 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 $abc$.

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

whether the following relation is${{ \frac{1}{{{x^{a - b}}}}} ^{\frac{1}{{a - c}}}}{{ \frac{1}{{{x^{b - c}}}}} ^{\frac{1}{{b - a}}}}{{ \frac{1}{{{x^{c - a}}}}} ^{^{\frac{1}{{c - b}}}}} = 1$

  1. True

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

Using exponent rules, each term simplifies to x^(b-a)/(a-c) * x^(c-b)/(b-a) * x^(a-c)/(c-b). Adding the exponents (b-a)/(a-c) + (c-b)/(b-a) + (a-c)/(c-b) results in 0, and x^0 = 1.

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

$(2^{0} + 4^{-1})\times 2^{2}$ is equal to

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

As we know that $a^{-b}$ is equal to $1/a^{b}$. Also, $p^{0}=1$ 


So, $(2^{0}+4^{-1})\times 2^{2}=(1+1/4)\times 2^{2}$ 

by using distributive law of multiplication , we get

 $(2^{0}+4^{-1})\times 2^{2}=1\times 2^{2}+1/4\times 2^{2}$ 

$(2^{0}+4^{-1})\times 2^2=4+1/4\times 4$ (because $2^{2}=2\times 2=4$) 

$(2^{0}+4^{-1})\times 2^{2}=4+1=5$

Multiple choice organization of commerce and management management by objectives (mbo) and management by exception (mbe) meaning and definition of mbo meaning and definition of mbe controlling

A cross relationship can be calculated using a formula of _______.

  1. n

  2. n(n-1)

  3. n(2^n-1 + n-1)

  4. None of the above
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

While the subordinates work under the same superior, they also interact amongst themselves. These are the relationships amongst subordinates known as cross relationship. The formula of cross relationship is n(n-1), where n= number of subordinates. 

Multiple choice

What is the value of the expression 2^3 + 3^3 + 4^3?

  1. 99

  2. 100

  3. 101

  4. 102
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

We can evaluate this expression using the formula for the sum of cubes: a^3 + b^3 + c^3 = (a + b + c)(a^2 + b^2 + c^2 - ab - bc - ca). Substituting a = 2, b = 3, and c = 4, we get 2^3 + 3^3 + 4^3 = (2 + 3 + 4)(2^2 + 3^2 + 4^2 - 2(3)(4) - 2(3)(4) - 2(4)(2)) = 9(4 + 9 + 16 - 24 - 24 - 16) = 9(100) = 100.

Multiple choice

What is the value of the expression 1^3 + 2^3 + 3^3 + ... + 10^3?

  1. 3025

  2. 3125

  3. 3225

  4. 3325
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The value of the expression 1^3 + 2^3 + 3^3 + ... + 10^3 is 3025.

Multiple choice

What is the value of the expression (2 + 3i)² - (2 - 3i)²?

  1. -16

  2. -12i

  3. 16

  4. 12i
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Using the formula (a + b)² = a² + 2ab + b², we can expand the expression as follows: (2 + 3i)² - (2 - 3i)² = (2² + 2 * 2 * 3i + (3i)²) - (2² - 2 * 2 * 3i + (3i)²) = (4 + 12i + 9i²) - (4 - 12i + 9i²) = 4 + 12i + 9i² - 4 + 12i - 9i² = 16.

Multiple choice

What is the value of the expression 2^3 + 3^2 - 4^1?

  1. 11

  2. 13

  3. 15

  4. 17
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

We can evaluate the expression by following the order of operations: 2^3 = 8, 3^2 = 9, 4^1 = 4. Therefore, the value of the expression is 8 + 9 - 4 = 15.

Multiple choice

What is the value of the expression (2 + 3i)(4 - 5i)?

  1. -11 + 22i

  2. -11 - 22i

  3. 11 + 22i

  4. 11 - 22i
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

We can evaluate the expression by using the distributive property: (2 + 3i)(4 - 5i) = 2(4 - 5i) + 3i(4 - 5i). Simplifying this expression, we get 8 - 10i + 12i - 15i^2. Since i^2 = -1, we can simplify further to get 11 - 22i.