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 general knowledge

What is called a polynomial with exactly 2 terms?

  1. binomial

  2. degree

  3. linear

  4. square
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

A polynomial with exactly two terms is called a binomial, such as x + 3 or 2a² - 5b. Degree refers to the highest power, linear refers to first-degree polynomials, and square is not a polynomial classification.

Multiple choice

To produce the addition of polynomials which of the following steps to be performed?

  1. The terms of the polynomials are scanned from left to right.

  2. The terms with powers that occurs only in one polynomials are simply copied into resulting polynomial.

  3. The coefficients of terms with same powers are added and then the new term is copied into the resulting polynomial.

  4. All of these

Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Adding polynomials represented as linked lists requires: scanning terms left to right, copying unique terms directly to the result, and combining coefficients of like-powered terms. All three operations are essential for complete polynomial addition, making 'All of these' the correct answer.

Multiple choice maths algebraic formulae - expansion of squares introduction to factorization introduction to factorisation factorising algebraic expressions

If the polynomial $f(x)$ is such that $f(-43) = 0$, which of the following is the factor of $f(x)$?

  1. $x - 43$
  2. $x$
  3. $x - 7$
  4. $x + 43$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

$f(-43)=0$ 
$\Rightarrow -43$ is root of $f(x)$

$\Rightarrow (x+43).g(x)=f(x)$ for some function $ g(x)$
$\Rightarrow (x+43)$ is the factor of $f(x)$

Multiple choice maths brackets order operations and algebra using brackets in algebraic expressions order of operations

If r and s are zeroes of the polynomial $t^2-4t+3$, then $\dfrac{1}{r}+\dfrac{1}{s}-2rs+\dfrac{14}{3}$ is equal to

  1. 0

  2. 1

  3. 2

  4. -1
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation
Quadratic Equation
If $\alpha$ and $\beta$ are the roots of the quadratic equation $ax^2+bx+c=0$
then $\alpha + \beta=\dfrac{-b}{a}$ and $\alpha\beta=\dfrac{c}{a}$

$f(t)=t^2-4t+3$
$r+s=4$
$rs=3$
Now,
$\dfrac{1}{r}+\dfrac{1}{s}-2rs+\dfrac{14}{3}=\dfrac{r+s}{rs}-2rs+\dfrac{14}{3}$

$=\dfrac{4}{3}-6+\dfrac{14}{3}=\dfrac{4-18+14}{3}=0$
Multiple choice reciprocal equations theory of equations maths

The equation $3x^4-5x^3+3x^2-4x+5=0$ is of the type

  1. Quadratic

  2. Linear

  3. Reiprocal

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

Given equation is $3x^4-5x^3+3x^2-4x+5=0$ .... $(i)$

The maximum power of $x$ in this equation is $ 4$, so this is $4th$ degree equation  
So, $(i)$ is neither linear nor quadratic equation.
In reciprocal equation ($ ax^4 +bx^3 +cx^2 +dx +e = 0 $)  the multiplication of the roots i.e ($\dfrac{e}{a}$) should be  $1$

In the given equation $(i)$, $\dfrac{e}{a}  = \dfrac{5}{3}$  
So the multiplication of roots is not equal to one
Therefore, equation $(i)$ is not a reciprocal equation.

Hence, option D is correct.

Multiple choice business maths functions and graphs some functions and their graphs -i graphs of the form y=ax^2+bx+c introduction to sets

let $f(x)$ be a polynomial of degree $4$ having extreme values at $x=2$.if $\underset { x\rightarrow 0 }{ lim } \left( \frac { f\left( x \right)  }{ { x }^{ 2 } } +1 \right) =3$ then $f(1)$

  1. $\frac { 1 }{ 2 } $
  2. $\frac { 3 }{ 2 } $
  3. $\frac { 5 }{ 2 }$
  4. $\frac { 9 }{ 2 } $
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Given f(x) is a quartic polynomial with extreme values at x=2, f'(2)=0. The limit condition lim(x->0) (f(x)/x^2 + 1) = 3 implies f(0)=0 and f'(0)=0, and f''(0)/2 = 2, so f''(0)=4. Using these conditions, one can determine the coefficients of f(x).

Multiple choice business maths functions and graphs some functions and their graphs -i graphs of the form y=ax^2+bx+c introduction to sets

Let $f(x)$ is cubic polynomial with real coefficient such that $f''(3) = 0, f'(5) = 0$. If $f(3) = 1$ and $f(5) = -3$, then $f(1)$ is equal to

  1. $2$
  2. $3$
  3. $5$
  4. $6$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Since f(x) is a cubic polynomial, its second derivative f''(x) is a linear function. Given f''(3) = 0, we can write f''(x) = c(x - 3). Integrating twice and using the conditions f'(5) = 0, f(3) = 1, and f(5) = -3 allows us to find the exact cubic polynomial and evaluate f(1) to be 5.