Practice Questions on Polynomials

Last Updated : 23 Jul, 2025

Polynomials are fundamental algebraic expressions that consist of variables and coefficients, incorporating the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables. Understanding polynomials is crucial for solving various mathematical problems in algebra and calculus.

What are Polynomials?

A polynomial is an expression of the form:

P(x)=aⁿxⁿ+aⁿ⁻¹xⁿ⁻¹+⋯+a¹x+a⁰

where aⁿ,aⁿ⁻¹,…,a¹,a⁰ are constants (coefficients) and x is a variable. The degree of the polynomial is the highest power of x that appears in the polynomial.

Important Formulas Related to Polynomials

Sum of Polynomials:

(aⁿxⁿ+⋯+a¹x+a⁰)+(bⁿxⁿ+⋯+b¹x+b⁰)=(aⁿ+bⁿ)xⁿ+⋯+(a¹+b¹)x+(a⁰+b⁰)

The sum of two polynomials P(x) and Q(x) is defined as the polynomial (P(x) + Q(x))(x), where each term of P(x) is added to the corresponding term of Q(x) with the same exponent.

Product of Polynomials:

(aⁿxⁿ+⋯+a¹x+a⁰)⋅(b^mx^m+⋯+b¹x+b⁰)(aⁿxⁿ+⋯+a¹x+a⁰)⋅(b^mx^m+⋯+b¹x+b⁰)

The product of two polynomials P(x) and Q(x) is defined as the polynomial (P(x) × Q(x))(x), where each term of P(x) is multiplied by the corresponding term of Q(x).

Derivative of a Polynomial:

d/dx(aⁿxⁿ+⋯+a¹x+a⁰)=naⁿxⁿ⁻¹+⋯+a¹

The derivative of a polynomial function P(x) with respect to x is another polynomial function denoted as P′(x). The derivative of P(x) with respect to x can be calculated using the formula P′(x) = (dP(x) / dx). The formula for calculating the derivative of a term with an exponent in a polynomial is (dP(x) / dx) = (coefficient × exponent). When the exponent is negative, the coefficient is negative and the exponent becomes positive (exponent - 1).

Read More: Polynomials

Solved Questions on Polynomials

Problem 1: Find the product of P(x)=x³−2x²+x−4 and Q(x)=2x+3

Solution:
P(x)⋅Q(x)=(x³−2x²+x−4)(2x+3)
=x³(2x+3)−2x²(2x+3)+x(2x+3)−4(2x+3)
=2x⁴+3x³−4x³−6x²+2x²+3x−8x−12
=2x⁴−x³−4x²−5x−12

Problem 2: Determine the sum of P(x)=5x²−3x+1 and Q(x)=−2x²+4x−6.

Solution:
P(x)+Q(x)=(5x²−3x+1)+(−2x²+4x−6)
=5x²−2x²−3x+4x+1−6
=3x²+x−5

Problem 3: Compute the derivative of P(x)=3x⁴−5x²+6x+8.

Solution:
P′(x)=d/dx(3x⁴−5x²+6x+8)
=12x³−10x+6

Problem 4: If P(x)=x³+x²−x+1 and Q(x)=2x³−3x+4, find P(x)−Q(x).

Solution:
P(x)−Q(x)=(x³+x²−x+1)−(2x³−3x+4)
=x³−2x³+x²−x+3x+1−4
=−x³+x²+2x−3

Problem 5: Evaluate P(x) at x=2 for P(x)=4x³−3x+5.

Solution:
P(2)=4(2)3−3(2)+5
=4⋅8−6+5
=32−6+5=31

Problem 6: Find the quotient and remainder when P(x)=x⁴−2x³+3x²−x+6 is divided by x−1.

Solution:
Using synthetic division:
Quotient: x3−x2+2x+1
Remainder: 7

Problem 7: If P(x)=x²+2x+1 and Q(x)=x+1, find P(x)/Q(x).

Solution:
P(x)/Q(x)=x²+2x+1 / x+1=(x+1)²/x+1 = x+1 (x≠−1)

Problem 8: Compute the integral of the polynomial P(x)=x³−2x²+x−4.

Solution:
∫P(x) dx=∫(x³−2x²+x−4) dx
=x⁴/4−2x³/3+x²/2−4x+C