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

Find the roots of the polynomial $x^5 - x^4 - 2x^3 + 2x^2 + x - 2 = 0$.

  1. $x = 1, -1, 2, -2, i$
  2. $x = 1, -1, 2, -2, -i$
  3. $x = 1, -1, 2, -2, \sqrt{2}$
  4. $x = 1, -1, 2, -2, -\sqrt{2}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

To find the roots of the polynomial, we can use the rational root theorem or Descartes' rule of signs. Using the rational root theorem, we find that $x = 1$ and $x = -1$ are roots. Dividing the polynomial by $x - 1$, we get $x^4 - 2x^3 - x^2 + x + 2$. Dividing this polynomial by $x + 1$, we get $x^3 - 3x^2 + x - 2$. Factoring this cubic, we get $(x - 2)(x^2 - x + 1)$. Solving the quadratic $x^2 - x + 1 = 0$, we get $x = \frac{1 \pm \sqrt{1 - 4(1)(1)}}{2(1)} = \frac{1 \pm \sqrt{-3}}{2}$. Therefore, the roots of the polynomial are $x = 1, -1, 2, -2, i$.

Multiple choice

What is a polynomial function?

  1. A function whose graph is a straight line

  2. A function whose graph is a parabola

  3. A function whose graph is a circle

  4. A function whose graph is a hyperbola
Reveal answer Fill a bubble to check yourself
Correct answer
Explanation

A polynomial function is a function whose graph is a polynomial curve.

Multiple choice

What is a polynomial function?

  1. A function whose graph is a polynomial curve

  2. A function whose equation is a polynomial equation

  3. A function whose rate of change is not constant

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

A polynomial function is a function whose graph is a polynomial curve, whose equation is a polynomial equation, and whose rate of change is not constant.

Multiple choice

What is the name of the famous mathematical problem that asks whether there is a way to find a solution to a polynomial equation of degree 5 or higher using only algebraic operations?

  1. The Goldbach conjecture

  2. The Riemann hypothesis

  3. The Fermat's Last Theorem

  4. The Abel-Ruffini theorem
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

The Abel-Ruffini theorem is one of the most important theorems in algebra, and it states that there is no general algebraic solution to a polynomial equation of degree 5 or higher.

Multiple choice

What is the Jones polynomial?

  1. A polynomial that is used to classify quantum knots.

  2. A polynomial that is used to classify classical knots.

  3. A polynomial that is used to classify links.

  4. A polynomial that is used to classify braids.
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The Jones polynomial is a polynomial that is used to classify quantum knots. It was discovered by Vaughan Jones in 1984, and it is one of the most important invariants of quantum knots. The Jones polynomial is a powerful tool for studying quantum knots, and it has been used to prove a number of important results about them.

Multiple choice

The dimension of the vector space of all polynomials of degree less than or equal to n is:

  1. n

  2. n+1

  3. n-1

  4. 2n
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The dimension of the vector space of all polynomials of degree less than or equal to n is n+1.

Multiple choice

Factor the polynomial (x^2 + 5x + 6).

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

  2. (x - 2)(x - 3)

  3. (x + 1)(x + 6)

  4. (x - 1)(x - 6)
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

To factor (x^2 + 5x + 6), find two numbers that add up to 5 and multiply to 6. These numbers are 2 and 3. Therefore, (x^2 + 5x + 6 = (x + 2)(x + 3)).

Multiple choice

Factor the polynomial (x^2 - 9).

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

  2. (x + 9)(x - 9)

  3. (x + 1)(x - 9)

  4. (x - 1)(x + 9)
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

To factor (x^2 - 9), recognize that it is a difference of squares. Therefore, (x^2 - 9 = (x + 3)(x - 3)).

Multiple choice

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

  1. (x + 2)^2

  2. (x - 2)^2

  3. (x + 4)(x - 4)

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

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

Multiple choice

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

  1. x(x + 1)^2

  2. x(x - 1)^2

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

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

To factor (x^3 + 2x^2 + x), factor out an (x) and then factor the remaining quadratic expression. Therefore, (x^3 + 2x^2 + x = x(x^2 + 2x + 1) = x(x + 1)^2).

Multiple choice

Factor the polynomial (x^3 - 8).

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

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

  3. (x - 2)^3

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

To factor (x^3 - 8), use the difference of cubes formula. Therefore, (x^3 - 8 = (x - 2)(x^2 + 2x + 4)).

Multiple choice

Factor the polynomial (x^4 - 16).

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

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

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

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

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

Multiple choice

Factor the polynomial (x^5 - 32).

  1. (x - 2)(x^4 + 2x^3 + 4x^2 + 8x + 16)

  2. (x + 2)(x^4 - 2x^3 + 4x^2 - 8x + 16)

  3. (x - 2)^5

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

To factor (x^5 - 32), use the difference of cubes formula. Therefore, (x^5 - 32 = (x - 2)(x^4 + 2x^3 + 4x^2 + 8x + 16)).

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)).