Mathematics ยท 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

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).

Multiple choice

What is the rational root theorem?

  1. Every rational root of a polynomial with integer coefficients is of the form \(p/q\), where \(p\) is a factor of the constant term and \(q\) is a factor of the leading coefficient.
  2. Every rational root of a polynomial with integer coefficients is of the form \(p/q\), where \(p\) is a factor of the leading coefficient and \(q\) is a factor of the constant term.
  3. Every rational root of a polynomial with integer coefficients is of the form \(p/q\), where \(p\) is a factor of the constant term and \(q\) is the leading coefficient.
  4. Every rational root of a polynomial with integer coefficients is of the form \(p/q\), where \(p\) is the leading coefficient and \(q\) is a factor of the constant term.
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The rational root theorem states that every rational root of a polynomial with integer coefficients is of the form (p/q), where (p) is a factor of the constant term and (q) is a factor of the leading coefficient.

Multiple choice

What is Descartes' rule of signs?

  1. The number of positive real roots of a polynomial is equal to the number of sign changes in the coefficients of the polynomial.

  2. The number of negative real roots of a polynomial is equal to the number of sign changes in the coefficients of the polynomial.

  3. The number of positive real roots of a polynomial is equal to the number of sign changes in the coefficients of the polynomial, plus the number of negative coefficients.

  4. The number of negative real roots of a polynomial is equal to the number of sign changes in the coefficients of the polynomial, plus the number of positive coefficients.
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Descartes' rule of signs states that the number of positive real roots of a polynomial is equal to the number of sign changes in the coefficients of the polynomial.

Multiple choice

Which mathematical concept is central to the study of polynomials?

  1. Variables

  2. Exponents

  3. Coefficients

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

The degree of a polynomial is a central concept in the study of polynomials, as it determines the highest power of the variable in the polynomial and influences its behavior.