Algebraic Polynomials & Identities Questions with Solutions

31 questions8 easy13 medium10 hard

A polynomial is an algebraic expression made of terms with non-negative whole-number powers of a variable. This topic covers the degree and zeros of a polynomial, the remainder and factor theorems, factorisation, and standard algebraic identities such as (a + b)^2, (a - b)^2, a^2 - b^2, (a + b)^3, a^3 + b^3 and a^3 + b^3 + c^3 - 3abc. Identities are among the biggest time-savers in aptitude. Questions that seem to need heavy calculation, such as evaluating x^3 + 1/x^3 when x + 1/x is known, become one-line answers when the right identity is used. The remainder theorem similarly lets you find a remainder without long division: the remainder when p(x) is divided by (x - a) is p(a). Practise the polynomial and identity questions below, each with a step-by-step solution.

All Questions Easy Medium Hard

Q1.
If the highest common factor of $x^{2} - p x - q$ and $5 x^{2} - 3 p x - 15 q$ is $(x - 3)$ , then the values of $p$ and $q$ will be
A. $\frac{5}{3}, 4$ B. $- \frac{5}{3}, - 4$ C. $\frac{5}{2}, \frac{3}{2}$ D. $4, - \frac{3}{5}$
hardView solution ›
Q2.
If $x^{2} + (4 - 3) x - 1 = 0$ , then what is the value of $x^{2} + \frac{1}{x ^{2}}$ ?
A. $9 - 83$ B. $21 - 123$ C. $21 - 83$ D. $17 - 83$
hardView solution ›
Q3.
If $x + \frac{1}{x} = 13$ , then one of the values of $x^{3} - \frac{1}{x ^{3}}$ is
A. $411$ B. $32$ C. $413$ D. $36$
hardView solution ›
Q4.
If $x^{3} + y^{3} = 468$ and $x + y = 12$ , then the value of $x^{4} + y^{4}$ will be
A. $3620$ B. $2036$ C. $3025$ D. $3026$
hardView solution ›
Q5.
If $a + b + c = 11$ , $ab + b c + c a = 3$ and $ab c = - 135$ , then what is the value of $a^{3} + b^{3} + c^{3}$ ?
A. $823$ B. $9293$ C. $925$ D. $827$
hardView solution ›
Q6.
If $x = 2 + 2^{2/3} + 2^{1/3}$ , then the value of the expression $x^{3} - 6 x^{2} + 6 x$ will be
A. $2$ B. $1$ C. $0$ D. $- 2$
hardAsked in MCA CET 2018View solution ›
Q7.
Let $x, y, z$ be positive real numbers such that $2 x + y - 3 y + z = 2$ and $4 x - 5 y - 9 z = 8$ . Then the value of $\frac{20 x + 38 y + 18 z + 1}{9 y + 9 z + 2}$ is:
A. 3B. 4C. 5D. 6
hardAsked in NIMCET 2026View solution ›
Q8.
The following polynomial has integer roots $x^{3} + 30 x^{2} - 7377 x + 14626 = 0$ . Find the value of the largest root.
A. 71B. 61C. 81D. 41
hardAsked in MAH MCA CET 2024View solution ›
Q9.
If x and y are positive real numbers satisfying the system of equations $x^{2} + y x y = 336$ and $y^{2} + x x y = 112$ , then x + y is:
A. $448$ B. $224$ C. 20D. 40
hardAsked in NIMCET 2014View solution ›
Q10.
Given the equations $x + y = 1$ , $x^{2} + y^{2} = 2$ , $x^{5} + y^{5} = A$ . Let $N$ be the number of solution pairs $(x, y)$ of this system. Then $A N$ is equal to
A. $\frac{7}{2}$ B. 3C. $\frac{19}{2}$ D. 2
hardAsked in NIMCET 2025View solution ›