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

Q21.
If $x + y + z = 0$ , then what is $(y + z - x)^{3} + (z + x - y)^{3} + (x + y - z)^{3}$ equal to?
A. $(x + y + z)^{3}$ B. $3 (x + y) (y + z) (z + x)$ C. $24 x y z$ D. $- 24 x y z$
mediumAsked in MCA CET 2022View solution ›
Q22.
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 ›
Q23.
HCF and LCM of two polynomials are $(x + 3)$ and $(x^{3} - 9 x^{2} - x + 105)$ respectively. If one of the two polynomials is $x^{2} - 4 x - 21$ , then the other is
A. $x^{2} + 2 x - 21$ B. $x^{2} + 2 x + 15$ C. $x^{2} - 2 x - 15$ D. $x^{2} - x - 15$
mediumAsked in MCA CET 2017View solution ›
Q24.
$(4 x^{3} y - 6 x^{2} y^{2} + 4 x y^{3} - y^{4})$ can be expressed as
A. $(x - y)^{4} - x^{4}$ B. $(x + y)^{4} - y^{4}$ C. $(x + y)^{4} - x^{4}$ D. $x^{4} - (x - y)^{4}$
mediumAsked in MCA CET 2016View solution ›
Q25.
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 ›
Q26.
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 ›
Q27.
If $a + b + c = 0$ , then the value of $\frac{a ^{2}}{b c} + \frac{b ^{2}}{c a} + \frac{c ^{2}}{ab}$ is
A. 1B. 0C. 3D. $- 1$
easyAsked in NIMCET 2020View solution ›
Q28.
For what value of $p$ is the polynomial $x^{4} - 3 x^{3} + 2 p x^{2} - 6$ exactly divisible by $(x - 1)$ ?
A. 2B. 4C. 6D. 8
easyAsked in NIMCET 2021View solution ›
Q29.
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 ›
Q30.
$a$ , $b$ , $c$ are positive integers such that $a^{2} + 2 b^{2} - 2 b c = 100$ and $2 ab - c^{2} = 100$ . Then the value of $\frac{a + b}{c}$ is
A. 10B. 100C. 2D. 20
mediumAsked in NIMCET 2015View solution ›
Q31.
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 ›

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