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
$(2^{0} + 4^{-1})\times 2^{2}$ is equal to
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 $2^{2}=2\times 2=4$)
$(2^{0}+4^{-1})\times 2^{2}=4+1=5$
A cross relationship can be calculated using a formula of _______.
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n
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n(n-1)
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n(2^n-1 + n-1)
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None of the above
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.
Subtract : $3x^{2}+2yx$ from $2x^{2}+3yx$
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$x^{2}-4xy$
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$5x^{2}y$
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$xy-x^{2}$
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$x^{2}+4xy$
C
Correct answer
Explanation
$(2x^{2}+3yx)-(3x^{2}+2yx)$
$=2x^{2}+3xy-3x^{2}-2xy$
$=-x^{2}+xy\Rightarrow (C)$
$Factor \times Factor$ is equal to ____ .
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difference
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multiple
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sum
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None of these
B
Correct answer
Explanation
$×$ sign between any $2$ numbers results in the product of those numbers or the multiple.
So, option B is the correct answer.
Find the minimal sum-of-products form of the following Boolean expression: (A ∨ B) ∧ (¬A ∨ C) ∧ (B ∨ ¬C)
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A ∨ B ∨ C
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A ∨ C
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B ∨ C
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A ∨ B
B
Correct answer
Explanation
Using Boolean algebra, we can simplify the expression as follows: (A ∨ B) ∧ (¬A ∨ C) ∧ (B ∨ ¬C) = (A ∨ B ∨ C) ∧ (¬A ∨ C) = (A ∨ C) ∧ (¬A ∨ C) = A ∨ C
Find the minimal sum-of-products form of the following Boolean expression: (A ∨ B) ∧ (¬A ∨ C) ∧ (B ∨ ¬C)
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A ∨ B ∨ C
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A ∨ C
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B ∨ C
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A ∨ B
B
Correct answer
Explanation
Using Boolean algebra, we can simplify the expression as follows: (A ∨ B) ∧ (¬A ∨ C) ∧ (B ∨ ¬C) = (A ∨ B ∨ C) ∧ (¬A ∨ C) = (A ∨ C) ∧ (¬A ∨ C) = A ∨ C
What is the value of the expression 2^3 + 3^3 + 4^3?
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.
What is the value of the expression 1^3 + 2^3 + 3^3 + ... + 10^3?
A
Correct answer
Explanation
The value of the expression 1^3 + 2^3 + 3^3 + ... + 10^3 is 3025.
What is the value of the expression (2 + 3i)² - (2 - 3i)²?
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.
Use the Möbius Inversion Formula to find a formula for the sum of the Möbius function over the divisors of an integer ( n ) that are perfect squares.
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\( \sum_{d|n} \mu(d) = 1 \)
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\( \sum_{d|n} \mu(d) = n \)
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\( \sum_{d|n} \mu(d) = \phi(n) \)
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\( \sum_{d|n} \mu(d) = \sigma(n) \)
Correct answer
Explanation
If ( n ) is a perfect square, then all of its prime factors appear with even exponents. Therefore, ( \mu(d) = 1 ) for all ( d | n ) that are perfect squares. Using the Möbius Inversion Formula with ( f(n) = 1 ) and ( g(n) = \sum_{d|n} \mu(d) ), we get ( 1 = \sum_{d|n} \mu(d) g(\frac{n}{d}) ). Since ( g(\frac{n}{d}) = 1 ) for all ( d | n ) that are perfect squares, we have ( \sum_{d|n} \mu(d) = \begin{cases} 1 & \text{if } n \text{ is a perfect square} \ 0 & \text{otherwise} \end{cases} ).
What is the value of the expression 2^3 + 3^2 - 4^1?
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.
What is the value of the expression (2 + 3i)(4 - 5i)?
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-11 + 22i
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-11 - 22i
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11 + 22i
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11 - 22i
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.
What is the multiplicative identity of a field?
B
Correct answer
Explanation
The multiplicative identity of a field is the element that, when multiplied by any other element of the field, results in that element. In the set of real numbers, the multiplicative identity is 1.
What is the name of the method used by Dandin to solve quadratic equations?
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Completing the Square
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Quadratic Formula
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Factorization
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Substitution
A
Correct answer
Explanation
Dandin used the method of completing the square to solve quadratic equations.
What is the product of (2x + 3) and (x - 4)?
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2x^2 - 5x - 12
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2x^2 + 5x - 12
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2x^2 - 5x + 12
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2x^2 + 5x + 12
A
Correct answer
Explanation
To multiply (2x + 3) and (x - 4), we use the distributive property. First, we multiply 2x by x and 2x by -4, which gives us 2x^2 and -8x. Then, we multiply 3 by x and 3 by -4, which gives us 3x and -12. Finally, we add the like terms to get 2x^2 - 5x - 12.