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

What is the value of the expression $2^3 + 3^2 - 5^1$?

  1. 14

  2. 16

  3. 18

  4. 20
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Using the order of operations, we have $2^3 + 3^2 - 5^1 = 8 + 9 - 5 = 18$.

Multiple choice

What is the name of the mathematical operation that finds the quotient of two numbers?

  1. Addition

  2. Subtraction

  3. Multiplication

  4. Division
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Division is a mathematical operation that finds the quotient of two numbers.

Multiple choice

Which of the following is a field extension of the field (ℚ, +, ×)?

  1. (ℝ, +, ×)

  2. (ℂ, +, ×)

  3. (ℚ(√2), +, ×)

  4. (ℤ/5ℤ, +, ×)
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

A field extension is a field that contains another field as a subfield. The field (ℚ(√2), +, ×) is a field extension of (ℚ, +, ×) because it contains (ℚ, +, ×) as a subfield.

Multiple choice

What is the degree of the field extension (ℚ(√2), +, ×) over (ℚ, +, ×)?

  1. 1

  2. 2

  3. 3

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

The degree of a field extension is the dimension of the extension field as a vector space over the base field. The degree of (ℚ(√2), +, ×) over (ℚ, +, ×) is 2 because (ℚ(√2), +, ×) is a two-dimensional vector space over (ℚ, +, ×).

Multiple choice

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

  1. 100

  2. 110

  3. 120

  4. 130
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

We can simplify the expression as follows: 2^3 + 3^3 + 4^3 - 3^2 - 4^2 = (2^3 + 3^3 + 4^3) - (3^2 + 4^2) = (8 + 27 + 64) - (9 + 16) = 99 - 25 = 110.

Multiple choice

Divide (x^3 - 2x^2 + x - 2) by (x - 2).

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

Using polynomial long division, we find that the quotient is (x^2 + 2x + 4) with a remainder of 0.

Multiple choice

What is the result of dividing (2x^3 + 3x^2 - 5x + 2) by (x + 1)?

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

Using synthetic division, we find that the quotient is (2x^2 - x + 2) with a remainder of 0.

Multiple choice

Divide (x^4 - 2x^3 + 3x^2 - 4x + 5) by (x - 1).

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

Using polynomial long division, we find that the quotient is (x^3 - x^2 + 2x - 3) with a remainder of 2.

Multiple choice

What is the result of dividing (3x^2 - 5x + 2) by (x - 2)?

  1. \(3x - 11\)
  2. \(3x + 11\)
  3. \(3x - 1\)
  4. \(3x + 1\)
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Using synthetic division, we find that the quotient is (3x - 1) with a remainder of 4.

Multiple choice

Divide (2x^3 + 5x^2 - 3x + 4) by (x + 2).

  1. \(2x^2 - x + 2\)
  2. \(2x^2 + x + 2\)
  3. \(2x^2 - x - 2\)
  4. \(2x^2 + x - 2\)
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Using polynomial long division, we find that the quotient is (2x^2 - x - 2) with a remainder of 0.

Multiple choice

Divide (x^3 - 3x^2 + 2x - 5) by (x - 1).

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

Using polynomial long division, we find that the quotient is (x^2 - 2x + 3) with a remainder of 2.

Multiple choice

What is the result of dividing (6x^2 - 11x + 3) by (3x - 1)?

  1. \(2x - 3\)
  2. \(2x + 3\)
  3. \(2x - 1\)
  4. \(2x + 1\)
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Using synthetic division, we find that the quotient is (2x - 3) with a remainder of 6.

Multiple choice

Divide (x^4 - 2x^3 + 3x^2 - 4x + 5) by (x - 1).

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

Using polynomial long division, we find that the quotient is (x^3 - x^2 + 2x - 3) with a remainder of 2.

Multiple choice

Divide (x^3 - 3x^2 + 2x - 5) by (x - 1).

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

Using polynomial long division, we find that the quotient is (x^2 - 2x + 3) with a remainder of 2.

Multiple choice

What is the result of dividing (6x^2 - 11x + 3) by (3x - 1)?

  1. \(2x - 3\)
  2. \(2x + 3\)
  3. \(2x - 1\)
  4. \(2x + 1\)
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Using synthetic division, we find that the quotient is (2x - 3) with a remainder of 6.