Mathematics ยท 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

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.

Multiple choice

What is the degree of a polynomial?

  1. The highest exponent of the variable in the polynomial.

  2. The lowest exponent of the variable in the polynomial.

  3. The number of terms in the polynomial.

  4. The number of variables in the polynomial.
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The degree of a polynomial is the highest exponent of the variable in the polynomial. For example, the polynomial (x^2 + 2x + 1) has a degree of 2, because the highest exponent of (x) is 2.

Multiple choice

What is the leading coefficient of a polynomial?

  1. The coefficient of the term with the highest degree.

  2. The coefficient of the term with the lowest degree.

  3. The coefficient of the first term in the polynomial.

  4. The coefficient of the last term in the polynomial.
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The leading coefficient of a polynomial is the coefficient of the term with the highest degree. For example, the polynomial (x^2 + 2x + 1) has a leading coefficient of 1, because the term with the highest degree is (x^2) and its coefficient is 1.

Multiple choice

What is the constant term of a polynomial?

  1. The term with the highest degree.

  2. The term with the lowest degree.

  3. The term with no variable.

  4. The term with the variable with the highest degree.
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The constant term of a polynomial is the term with no variable. For example, the polynomial (x^2 + 2x + 1) has a constant term of 1, because the term with no variable is 1.

Multiple choice

What is the sum of two polynomials?

  1. The polynomial obtained by adding the coefficients of the corresponding terms.

  2. The polynomial obtained by subtracting the coefficients of the corresponding terms.

  3. The polynomial obtained by multiplying the coefficients of the corresponding terms.

  4. The polynomial obtained by dividing the coefficients of the corresponding terms.
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The sum of two polynomials is the polynomial obtained by adding the coefficients of the corresponding terms. For example, the sum of the polynomials (x^2 + 2x + 1) and (x^2 - x + 2) is (2x^2 + x + 3).

Multiple choice

What is the difference of two polynomials?

  1. The polynomial obtained by adding the coefficients of the corresponding terms.

  2. The polynomial obtained by subtracting the coefficients of the corresponding terms.

  3. The polynomial obtained by multiplying the coefficients of the corresponding terms.

  4. The polynomial obtained by dividing the coefficients of the corresponding terms.
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The difference of two polynomials is the polynomial obtained by subtracting the coefficients of the corresponding terms. For example, the difference of the polynomials (x^2 + 2x + 1) and (x^2 - x + 2) is (2x + 1).

Multiple choice

What is the product of two polynomials?

  1. The polynomial obtained by adding the coefficients of the corresponding terms.

  2. The polynomial obtained by subtracting the coefficients of the corresponding terms.

  3. The polynomial obtained by multiplying the coefficients of the corresponding terms.

  4. The polynomial obtained by dividing the coefficients of the corresponding terms.
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The product of two polynomials is the polynomial obtained by multiplying the coefficients of the corresponding terms. For example, the product of the polynomials (x^2 + 2x + 1) and (x^2 - x + 2) is (x^4 + x^3 - x^2 + 2x^2 + 4x + 2).

Multiple choice

What is the quotient of two polynomials?

  1. The polynomial obtained by adding the coefficients of the corresponding terms.

  2. The polynomial obtained by subtracting the coefficients of the corresponding terms.

  3. The polynomial obtained by multiplying the coefficients of the corresponding terms.

  4. The polynomial obtained by dividing the coefficients of the corresponding terms.
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

The quotient of two polynomials is the polynomial obtained by dividing the coefficients of the corresponding terms. For example, the quotient of the polynomials (x^2 + 2x + 1) and (x^2 - x + 2) is (x + 3).

Multiple choice

What is the remainder of two polynomials?

  1. The polynomial obtained by adding the coefficients of the corresponding terms.

  2. The polynomial obtained by subtracting the coefficients of the corresponding terms.

  3. The polynomial obtained by multiplying the coefficients of the corresponding terms.

  4. The polynomial obtained by dividing the coefficients of the corresponding terms.
Reveal answer Fill a bubble to check yourself
Correct answer
Explanation

The remainder of two polynomials is the polynomial obtained by subtracting the product of the divisor and the quotient from the dividend. For example, the remainder of the polynomials (x^2 + 2x + 1) and (x^2 - x + 2) is (-x + 1).

Multiple choice

What is the factor theorem?

  1. If \(x - a\) is a factor of a polynomial \(f(x)\), then \(f(a) = 0\).
  2. If \(x - a\) is a factor of a polynomial \(f(x)\), then \(f(a) \neq 0\).
  3. If \(x - a\) is not a factor of a polynomial \(f(x)\), then \(f(a) = 0\).
  4. If \(x - a\) is not a factor of a polynomial \(f(x)\), then \(f(a) \neq 0\).
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The factor theorem states that if (x - a) is a factor of a polynomial (f(x)), then (f(a) = 0).