Mathematics · Quantitative Aptitude

Algebraic Polynomials

132 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^{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 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.

Multiple choice

Which of the following is a property of the Legendre polynomials?

  1. They are orthogonal on the interval \([-1, 1]\)
  2. They satisfy the differential equation \((1-x^2)y'' - 2xy' + n(n+1)y = 0\)
  3. They are complete on the interval \([-1, 1]\)

  4. All of the above
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Legendre polynomials possess all of the mentioned properties.

Multiple choice

Which of the following is a property of the Legendre polynomials?

  1. They are orthogonal on the interval \([0, 1]\)
  2. They satisfy the differential equation \((1-x^2)y'' - 2xy' + n(n+1)y = 0\)
  3. They are complete on the interval \([0, 1]\)

  4. All of the above
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Legendre polynomials possess all of the mentioned properties.

Multiple choice

What is the solvability of a polynomial?

  1. The ability to express its roots in terms of radicals.

  2. The ability to express its roots in terms of elementary functions.

  3. The ability to express its roots in terms of algebraic functions.

  4. The ability to express its roots in terms of transcendental functions.
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The solvability of a polynomial is the ability to express its roots in terms of radicals.

Multiple choice

What is the irreducibility of a polynomial?

  1. The inability to factor the polynomial into a product of two non-constant polynomials.

  2. The inability to factor the polynomial into a product of two non-linear polynomials.

  3. The inability to factor the polynomial into a product of two non-quadratic polynomials.

  4. The inability to factor the polynomial into a product of two non-cubic polynomials.
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The irreducibility of a polynomial is the inability to factor the polynomial into a product of two non-constant polynomials.

Multiple choice

What is the Eisenstein criterion for irreducibility?

  1. If a polynomial has an integer coefficient, a leading coefficient of 1, and a constant term that is not divisible by the leading coefficient, then it is irreducible over the integers.

  2. If a polynomial has a rational coefficient, a leading coefficient of 1, and a constant term that is not divisible by the leading coefficient, then it is irreducible over the rationals.

  3. If a polynomial has a real coefficient, a leading coefficient of 1, and a constant term that is not divisible by the leading coefficient, then it is irreducible over the reals.

  4. If a polynomial has a complex coefficient, a leading coefficient of 1, and a constant term that is not divisible by the leading coefficient, then it is irreducible over the complex numbers.
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The Eisenstein criterion for irreducibility is a test for determining whether a polynomial with integer coefficients is irreducible over the integers.

Multiple choice

What is a polynomial?

  1. An expression consisting of variables and constants, combined using addition, subtraction, and multiplication.

  2. An expression consisting of variables and constants, combined using addition, subtraction, multiplication, and division.

  3. An expression consisting of variables and constants, combined using addition and subtraction.

  4. An expression consisting of variables and constants, combined using multiplication and division.
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

A polynomial is a mathematical expression consisting of variables and constants, combined using addition, subtraction, and multiplication. The variables represent unknown values, while the constants are fixed values.