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 maths remainder and factor theorems linear and synthetic method of division factorising expressions using factor theorem division algorithm for polynomials

When a polynomial $P(x)$ is divided by $x, (x - 2)$ and $(x - 3)$, remainders are $1$, $3$ and $2$ respectively. the same polynomial is divided by $x(x - 2)(x - 3)$, the remainder is $ax^2 + bx + c$, then the value of $c$ is

  1. $-3$
  2. $-2$
  3. $6$
  4. $1$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

According to factor theorem

$p(x)=xq(x)+1$
$P(x)=(x-2)q'(x)+3$
$P(x)=(x-3)q''(x)+2$
So, At $x=0 , P(0)=1$
At $x=2 , P(2)=3$
At $x=3 , P(3)=2$
Now when the polynomial $P(x)$ is divided by $(x-2)(x-3)x$ the remainder must have the degree less than $3$ . that is the remainder will be of the form $ax^2+bx+c$
$\implies P(x)=x(x-2)(x-3)q''''(x)+ax^2+bx+c$
So, $P(0) =1=a(0)^2+b(0)+c\implies c=1$

Multiple choice maths remainder and factor theorems linear and synthetic method of division factorising expressions using factor theorem division algorithm for polynomials

Which of the following given options is/ are correct?
If $p(x)=q(x)g(x)+r(x)$ (By Division Algorithm) where p(x), g(x) are any two polynomials with $g(x)\neq 0$, then

  1. $r(x)=0$ always

  2. degree of r(x)< degree of g(x) always
  3. either $r(x)=0$ or degree of r(x)< degree of g(x)
  4. $r(x)=g(x)$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

(a) If p(x) is not divisible by g(x), then $r(x)\neq 0 \therefore$ (a) is not true
(b) If p(x) is divisible by g(x), the $r(x)=0$ for all x i.e., r(x) is zero polynomial whose degree is not defined.
$\therefore $(b) is not true
(c) is clearly true [$\because$ division algorithm rule]
Since degree of $r(x)<$ degree of g(x) or $r(x)=0$, but $g(x)\neq 0$.
(d) $\therefore r(x)=g(x)$ is not true.

Multiple choice maths remainder and factor theorems linear and synthetic method of division factorising expressions using factor theorem division algorithm for polynomials

A polynomial $f(x)$ with rational coefficient leaves reminder $15$, when divided by $(x-3)$ and remainder $2x+1$, when divided by $(x-1)^{2}$. If $p$ is coefficient of $x$ of its remainder which will come out if $f(x)$ is divided by $(x-3)(x-1)^{2}$ then find $p$.

  1. $-2$
  2. $-1$
  3. $1$
  4. $6$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Let the remainder be ax^2+bx+c. Using f(3)=15 and the condition for (x-1)^2, we set up a system of equations. Solving for the remainder when divided by (x-3)(x-1)^2 yields a linear term coefficient p = -2.

Multiple choice maths multiplication and division of algebraic expressions linear and synthetic method of division factorising expressions using factor theorem division algorithm for polynomials

The polynomial $f(x)={ x }^{ 4 }-2{ x }^{ 3 }+3{ x }^{ 2 }-ax+b$ when divided by $(x-1)$ and $(x+1)$ leaves the remainders $5$ and $19$ respectively. Find the values of $a$ and $b$. Hence, find the remainder when $f(x)$ is divided by $(x-2)$

  1. $a=6,b=8$ and $remainder=10$
  2. $a=5,b=8$ and $remainder=10$
  3. $a=5,b=7$ and $remainder=10$

  4. None of these
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

    by remainder theorem,

      $f(1)=5$ $and$ $f(-1)=14$

     $\therefore 1-2+3-a+b=5\Rightarrow b-a=3$  and
     $1+2+3+a+b=19\Rightarrow a+b=13$ 
      $b=8$ $a=5$ 
       and remainder when $f(x)$ is divided by $(x-2)$ is
         $f(2)=16-16+12-10+8=10$

Multiple choice

What are the applications of the quadratic formula?

  1. Solving quadratic equations

  2. Finding the roots of polynomials

  3. Factoring polynomials
Reveal answer Fill a bubble to check yourself
Correct answer
Explanation

The quadratic formula has a wide range of applications, including solving quadratic equations, finding the roots of polynomials, and factoring polynomials.

Multiple choice

What is the minimal polynomial of a matrix?

  1. A polynomial that is the lowest-degree polynomial that annihilates the matrix

  2. A polynomial that is the lowest-degree polynomial that has the matrix as a root

  3. A polynomial that is the lowest-degree polynomial that has the eigenvalues of the matrix as its roots

  4. A polynomial that is the lowest-degree polynomial that has the eigenvectors of the matrix as its roots
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The minimal polynomial of a matrix is a polynomial that is the lowest-degree polynomial that annihilates the matrix.

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

Find two numbers that add up to 5 and multiply to 6. These numbers are 2 and 3. Rewrite the polynomial as (x^2 + 2x + 3x + 6). Group the terms: ((x^2 + 2x) + (3x + 6)). Factor each group: (x(x + 2) + 3(x + 2)). Combine the factors: ((x + 2)(x + 3)).

Multiple choice

Factor the polynomial: (x^2 - 9)

  1. \((x + 3)(x - 3)\)
  2. \((x + 1)(x - 9)\)
  3. \((x - 3)(x + 3)\)
  4. \((x - 1)(x + 9)\)
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

This is a difference of squares. Rewrite the polynomial as (x^2 - 3^2). Use the formula (a^2 - b^2 = (a + b)(a - b)). Substitute (a = x) and (b = 3): (x^2 - 3^2 = (x + 3)(x - 3)).

Multiple choice

Factor the polynomial: (x^2 + 7x + 12)

  1. \((x + 3)(x + 4)\)
  2. \((x + 2)(x + 6)\)
  3. \((x + 1)(x + 12)\)
  4. \((x - 3)(x - 4)\)
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Find two numbers that add up to 7 and multiply to 12. These numbers are 3 and 4. Rewrite the polynomial as (x^2 + 3x + 4x + 12). Group the terms: ((x^2 + 3x) + (4x + 12)). Factor each group: (x(x + 3) + 4(x + 3)). Combine the factors: ((x + 3)(x + 4)).

Multiple choice

Factor the polynomial: (x^2 - 4x - 21)

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

Find two numbers that add up to -4 and multiply to -21. These numbers are -7 and 3. Rewrite the polynomial as (x^2 - 7x + 3x - 21). Group the terms: ((x^2 - 7x) + (3x - 21)). Factor each group: (x(x - 7) + 3(x - 7)). Combine the factors: ((x + 3)(x - 7)).

Multiple choice

What is a continued fraction?

  1. A fraction where the denominator is a polynomial.

  2. A fraction where the numerator is a polynomial.

  3. A fraction where both the numerator and denominator are polynomials.

  4. None of the above
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

A continued fraction is a fraction where both the numerator and denominator are polynomials.

Multiple choice

What is the degree of the zero polynomial?

  1. 0

  2. 1

  3. 2

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

The degree of a polynomial is the highest exponent of the variable that appears in the polynomial. Since the zero polynomial has no variable, its degree is undefined.

Multiple choice

What is the Hilbert polynomial of the ring $\mathbb{C}[x, y]/(x^2 + y^2)$?

  1. $t^2$
  2. $t^3$
  3. $t^4$
  4. $t^5$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The Hilbert polynomial of a ring is a polynomial that gives the number of generators of the ring in each degree. In this case, the ring $\mathbb{C}[x, y]/(x^2 + y^2)$ is generated by the elements $x$ and $y$, so its Hilbert polynomial is $t^2$.

Multiple choice

Find the roots of the polynomial $x^3 - 2x^2 - 5x + 6 = 0$.

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

To find the roots of the polynomial, we can use synthetic division or the rational root theorem. Using synthetic division, we find that $x = 1$ is a root. Dividing the polynomial by $x - 1$, we get $x^2 - x - 6$. Factoring this quadratic, we get $(x - 3)(x + 2)$. Therefore, the roots of the polynomial are $x = 1, 2, 3$.

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 $x = 1, -1, 2, -2, i$.