Mathematics · Quantitative Aptitude

Algebraic Polynomials

138 Questions

Algebraic polynomials involve factoring expressions, applying the remainder theorem, and finding variable roots. These topics form a major part of the quantitative aptitude syllabus. Regular practice ensures accuracy in solving complex algebraic equations.

Factoring polynomialsFactor theoremRemainder theoremPolynomial rootsCubic polynomials

Algebraic Polynomials Questions

Multiple choice

Factor the polynomial (x^6 - 64).

  1. (x^3 + 4)(x^3 - 16)

  2. (x^3 + 8)(x^3 - 8)

  3. (x^2 + 8)(x^4 - 8x^2 + 64)

  4. (x^2 - 8)(x^4 + 8x^2 + 64)
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

To factor (x^6 - 64), recognize that it is a difference of cubes. Therefore, (x^6 - 64 = (x^3 + 4)(x^3 - 16) = (x^3 + 4)(x + 2)(x^2 - 2x + 4)).

Multiple choice

Factor the polynomial (x^8 - 1).

  1. (x^4 + 1)(x^4 - 1)

  2. (x^2 + 1)(x^6 - x^4 + x^2 - 1)

  3. (x^2 - 1)(x^6 + x^4 + x^2 + 1)

  4. (x + 1)(x^7 - x^6 + x^5 - x^4 + x^3 - x^2 + x - 1)
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

To factor (x^8 - 1), recognize that it is a difference of squares. Therefore, (x^8 - 1 = (x^4 + 1)(x^4 - 1) = (x^4 + 1)(x^2 + 1)(x^2 - 1)).

Multiple choice

Factor the polynomial (x^{10} - 1).

  1. (x^5 + 1)(x^5 - 1)

  2. (x^2 + 1)(x^8 - x^6 + x^4 - x^2 + 1)

  3. (x^2 - 1)(x^8 + x^6 + x^4 + x^2 + 1)

  4. (x + 1)(x^9 - x^8 + x^7 - x^6 + x^5 - x^4 + x^3 - x^2 + x - 1)
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

To factor (x^{10} - 1), recognize that it is a difference of squares. Therefore, (x^{10} - 1 = (x^5 + 1)(x^5 - 1) = (x^5 + 1)(x + 1)(x^4 - x^3 + x^2 - x + 1)).

Multiple choice

Factor the polynomial (x^3 + 3x^2 + 3x + 1).

  1. (x + 1)^3

  2. (x - 1)^3

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

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

To factor (x^3 + 3x^2 + 3x + 1), recognize that it is a sum of cubes. Therefore, (x^3 + 3x^2 + 3x + 1 = (x + 1)^3).

Multiple choice

Factor the polynomial (x^3 - 3x^2 + 3x - 1).

  1. (x - 1)^3

  2. (x + 1)^3

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

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

To factor (x^3 - 3x^2 + 3x - 1), recognize that it is a difference of cubes. Therefore, (x^3 - 3x^2 + 3x - 1 = (x - 1)^3).

Multiple choice

Factor the polynomial (x^4 + 6x^2 + 9).

  1. (x^2 + 3)^2

  2. (x^2 - 3)^2

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

  4. (x^2 + 9)(x^2 - 9)
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

To factor (x^4 + 6x^2 + 9), recognize that it is a perfect square trinomial. Therefore, (x^4 + 6x^2 + 9 = (x^2 + 3)^2).

Multiple choice

Factor the polynomial (x^4 - 6x^2 + 9).

  1. (x^2 + 3)^2

  2. (x^2 - 3)^2

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

  4. (x^2 + 9)(x^2 - 9)
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

To factor (x^4 - 6x^2 + 9), recognize that it is a perfect square trinomial. Therefore, (x^4 - 6x^2 + 9 = (x^2 - 3)^2).

Multiple choice

Which of the following sets of vectors is linearly independent in the vector space of polynomials of degree 2 or less?

  1. {(1, 0, 0), (0, 1, 0), (0, 0, 1)}

  2. {(1, 1, 0), (1, 0, 1), (0, 1, 1)}

  3. {(1, 2, 3), (4, 5, 6), (7, 8, 9)}

  4. {(1, 0, 1), (0, 1, 1), (1, 1, 1)}
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The set of vectors {(1, 0, 0), (0, 1, 0), (0, 0, 1)} is linearly independent because no vector in the set can be expressed as a linear combination of the other vectors.

Multiple choice

Consider the following set of vectors in the vector space of polynomials of degree 3 or less:

  1. {(1, 0, 0, 0), (0, 1, 0, 0), (0, 0, 1, 0), (0, 0, 0, 1)}

  2. {(1, 1, 1, 1), (1, 0, 1, 0), (1, 0, 0, 1), (0, 1, 0, 1)}

  3. {(1, 2, 3, 4), (5, 6, 7, 8), (9, 10, 11, 12)}

  4. {(1, 0, 1, 1), (0, 1, 1, 1), (1, 1, 1, 1)}
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The set of vectors {(1, 0, 0, 0), (0, 1, 0, 0), (0, 0, 1, 0), (0, 0, 0, 1)} is linearly independent because no vector in the set can be expressed as a linear combination of the other vectors.

Multiple choice

Consider the following set of vectors in the vector space of polynomials of degree 3 or less:

  1. {(1, 0, 0, 0), (0, 1, 0, 0), (0, 0, 1, 0), (0, 0, 0, 1)}

  2. {(1, 1, 1, 1), (1, 0, 1, 0), (1, 0, 0, 1), (0, 1, 0, 1)}

  3. {(1, 2, 3, 4), (5, 6, 7, 8), (9, 10, 11, 12)}

  4. {(1, 0, 1, 1), (0, 1, 1, 1), (1, 1, 1, 1)}
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The set of vectors {(1, 0, 0, 0), (0, 1, 0, 0), (0, 0, 1, 0), (0, 0, 0, 1)} is linearly independent because no vector in the set can be expressed as a linear combination of the other vectors.

Multiple choice

Consider the following set of vectors in the vector space of polynomials of degree 3 or less:

  1. {(1, 0, 0, 0), (0, 1, 0, 0), (0, 0, 1, 0), (0, 0, 0, 1)}

  2. {(1, 1, 1, 1), (1, 0, 1, 0), (1, 0, 0, 1), (0, 1, 0, 1)}

  3. {(1, 2, 3, 4), (5, 6, 7, 8), (9, 10, 11, 12)}

  4. {(1, 0, 1, 1), (0, 1, 1, 1), (1, 1, 1, 1)}
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The set of vectors {(1, 0, 0, 0), (0, 1, 0, 0), (0, 0, 1, 0), (0, 0, 0, 1)} is linearly independent because no vector in the set can be expressed as a linear combination of the other vectors.

Multiple choice

Which of the following is a polynomial ring over the field (ℤ/2ℤ, +, ×)?

  1. (ℤ/2ℤ[x], +, ×)

  2. (ℤ/3ℤ[x], +, ×)

  3. (ℤ/4ℤ[x], +, ×)

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

A polynomial ring over a field is the set of all polynomials with coefficients in that field. The polynomial ring (ℤ/2ℤ[x], +, ×) consists of all polynomials with coefficients in the field (ℤ/2ℤ, +, ×).

Multiple choice

What is the degree of the polynomial x^3 + x + 1 in the polynomial ring (ℤ/2ℤ[x], +, ×)?

  1. 1

  2. 2

  3. 3

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

The degree of a polynomial is the highest exponent of the variable in the polynomial. The degree of x^3 + x + 1 in (ℤ/2ℤ[x], +, ×) is 3.

Multiple choice

Which of the following is a maximal ideal of the polynomial ring (ℤ/2ℤ[x], +, ×)?

  1. (xℤ/2ℤ[x], +, ×)

  2. (x^2ℤ/2ℤ[x], +, ×)

  3. (x^3ℤ/2ℤ[x], +, ×)

  4. (x^4ℤ/2ℤ[x], +, ×)
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

A maximal ideal of a polynomial ring is a proper ideal that is not contained in any other proper ideal. The ideal (xℤ/2ℤ[x], +, ×) is a maximal ideal of (ℤ/2ℤ[x], +, ×) because it is a proper ideal and it is not contained in any other proper ideal.

Multiple choice

What is the fundamental theorem of algebra?

  1. Every polynomial equation of degree n has exactly n roots.

  2. Every polynomial equation of degree n has at least one root.

  3. Every polynomial equation of degree n has at most one root.

  4. Every polynomial equation of degree n has exactly n distinct roots.
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The fundamental theorem of algebra states that every polynomial equation of degree n has exactly n roots. This theorem is a fundamental result in algebra and has important implications for the study of polynomials and their applications.