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.
The remainder, when $3 x^{3} - 2 x^{2} y - 13 x y^{2} + 10 y^{3}$ is divided by $(x - 2 y)$ , is equal to
A. $0$ B. $y$ C. $y - 5$ D. $y + 3$
mediumView solution ›
Q2.
If $x = 2 - p$ , then $x^{3} + 6 x p + p^{3}$ is equal to
A. $12$ B. $6$ C. $8$ D. $4$
mediumView solution ›
Q3.
The value of $\frac{( x - y ) ^{3} + ( y - z ) ^{3} + ( z - x ) ^{3}}{9 ( x - y ) ( y - z ) ( z - x )}$ is
A. $0$ B. $\frac{1}{9}$ C. $\frac{1}{3}$ D. $1$
mediumView solution ›
Q4.
What is the value of $8 x^{3} + 18 x y + y^{3} - 27$ , when $2 x + y - 3 = 0$ ?
A. $- 27$ B. $27$ C. $0$ D. $1$
mediumView solution ›
Q5.
What is the square root of $(x^{2} - 14 x + 49) (x^{2} + 6 x + 9)$ ?
A. $(x - 4) (x + 9)$ B. $(x - 7) (x + 3)$ C. $(x - 1) (x + 17)$ D. $(x - 3) (x + 8)$
mediumView solution ›
Q6.
If $x + y = 4$ and $\frac{1}{x} + \frac{1}{y} = \frac{16}{15}$ , then what is the value of $(x^{3} + y^{3})$ ?
A. $18$ B. $19$ C. $21$ D. $16$
mediumView solution ›
Q7.
If $x, y, z$ are positive integers such that $x : y = 1 : 2$ and $y : z = 3 : 5$ , then which one of the following will be a possible value of $5 x - 3 y + z$ ?
A. $180$ B. $190$ C. $196$ D. $200$
mediumView solution ›
Q8.
What is the simplified value of $\frac{1}{8} {(x + \frac{1}{y})^{2} - (x - \frac{1}{y})^{2}}$ ?
A. $\frac{x}{2 y}$ B. $\frac{x}{y}$ C. $\frac{4 x}{y}$ D. $\frac{2 x}{y}$
mediumView solution ›
Q9.
If $(3 x - 1)^{3} + (4 x - 3)^{3} + (2 x + 1)^{3} = 3 (3 x - 1) (4 x - 3) (2 x + 1)$ and $x \neq = \frac{1}{3}$ , then $x = ?$
A. $\frac{1}{2}$ B. $2$ C. $\frac{1}{4}$ D. $1$
mediumView solution ›
Q10.
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 ›
Q11.
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 ›
Q12.
$(4 x^{3} y - 6 x^{2} y^{2} + 4 x y^{3} - y^{4})$ 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 ›
Q13.
$a$ , $b$ , $c$ 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 ›