CBSE Class 9 Maths Chapter 2 Polynomials Notes

Key points

• A polynomial in one variable x is an expression of the form a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0, where powers of x are whole numbers and coefficients are real numbers.
• The degree of a non-zero polynomial is the highest power of the variable with a non-zero coefficient.
• A constant non-zero polynomial has degree 0; the zero polynomial has no defined degree.
• If p(a)=0, then a is a zero of p(x), and x-a is a factor of p(x).
• Remainder theorem: when p(x) is divided by x-a, the remainder is p(a).
• Factor theorem: x-a is a factor of p(x) if and only if p(a)=0.
• Common degrees: linear = 1, quadratic = 2, cubic = 3.
• Main identities: (x+y)^2=x^2+2xy+y^2, (x-y)^2=x^2-2xy+y^2, x^2-y^2=(x-y)(x+y), (x+y)^3=x^3+y^3+3xy(x+y).
• For factorisation, first look for common factors, then identities, then factor theorem if a simple zero is visible.
• Always substitute with brackets, especially for negative values like p(-2).

Important definitions

Polynomial

An algebraic expression in which the variable has only non-negative integral powers. Example: 3x^2-5x+7.

Term

Each part of a polynomial separated by + or - signs. In 2x^2-3x+4, the terms are 2x^2, -3x and 4.

Coefficient

The numerical factor multiplying a variable term. In -7x^3, the coefficient of x^3 is -7.

Degree

The highest exponent of the variable in a non-zero polynomial. Degree of 5x^4-2x+1 is 4.

Constant polynomial

A polynomial with degree 0, such as 6 or -11, because it has no variable term.

Zero of a polynomial

A number a for which p(a)=0. It is also called a root of the polynomial.

Remainder theorem

If p(x) is divided by x-a, the remainder is p(a).

Factor theorem

x-a is a factor of p(x) exactly when p(a)=0.

Core concept: how to read a polynomial

A polynomial is built from constants, variables and whole-number powers. For Class 9, most questions use one variable, usually x. To identify the degree, ignore zero-coefficient terms and pick the highest power actually present. For example, in 4x^5-3x^2+8, the degree is 5. In 0x^7+2x^3-1, the degree is not 7 because 0x^7 is not an actual term; the degree is 3. To find a zero, substitute a value into the polynomial and check whether the result becomes 0. Diagram-friendly idea: draw the graph of y=p(x); wherever it cuts or touches the x-axis, p(x)=0, so that x-coordinate is a zero. In exams, you usually do not need to draw graphs, but this picture helps you remember why p(a)=0 is connected to factors like x-a.

Types of polynomials by degree and terms

Formula bank: identities to memorise

How to use remainder theorem and factor theorem

1. Identify the divisor — Write the divisor in the form x-a. If the divisor is x-3, then a=3. If it is x+2, rewrite as x-(-2), so a=-2.
2. Substitute the value — Find p(a) by replacing every x with a. Use brackets for negative values and powers.
3. Interpret the result — If p(a) is any number R, then R is the remainder when p(x) is divided by x-a.
4. Check factor condition — If p(a)=0, then x-a is a factor of p(x). If p(a) is not 0, x-a is not a factor.
5. Use it for factorisation — After finding one factor x-a, divide or factor the remaining expression to write p(x) completely as a product.

Mini worked examples

• {"title":"Find the degree","bodyMarkdown":"For p(x)=5x^3-2x^5+7x-1, the highest power is 5. So, degree = 5."}
• {"title":"Use remainder theorem","bodyMarkdown":"Find the remainder when p(x)=x^3-4x+1 is divided by x-2. Here a=2. p(2)=8-8+1=1. Remainder = 1."}
• {"title":"Check factor theorem","bodyMarkdown":"Check whether x+1 is a factor of p(x)=x^3+2x^2-x-2. Here a=-1. p(-1)=-1+2+1-2=0. Hence x+1 is a factor."}

Common mistakes to avoid

• Writing degree of the zero polynomial as 0. Correct: degree of zero polynomial is not defined.
• Treating 1/x, x^{-2}, or sqrt(x) as polynomial terms. They are not polynomial terms because powers are not non-negative integers.
• For x+3, using a=3 instead of a=-3. Always convert x+3 to x-(-3).
• For p(-2), forgetting brackets: (-2)^2=4 but -2^2=-4 if brackets are missing.
• Applying a^2+b^2=(a+b)^2. Correct: (a+b)^2=a^2+2ab+b^2.
• In factor theorem, saying x+a is a factor when p(a)=0. Correct: x-a is a factor when p(a)=0; x+a is a factor when p(-a)=0.

Exam tips and marking cues

For 1-mark questions, write only the required result clearly: degree, type, zero or remainder. For 2-3 mark questions, show substitution step like p(2)=... because marks are often given for using the theorem correctly. In factor theorem questions, do not just write yes/no; state p(a)=0 and then conclude that x-a is a factor. For identity-based factorisation, mention the identity pattern if the expression is long. Before finalising, re-expand mentally to check signs, especially in a^3-b^3 and (a-b)^3.

Quick revision checks

• Q: Is 4x^2 + 3/x - 1 a polynomial? A: No. The term 3/x = 3x^{-1} has a negative power.
• Q: What is the degree of 9x^4 - x^2 + 11? A: 4, because the highest power of x is 4.
• Q: Find the remainder when x^2 + 5x + 6 is divided by x + 2. A: Put x=-2. p(-2)=4-10+6=0. Remainder = 0.
• Q: If p(3)=0, which linear expression is a factor of p(x)? A: x-3 is a factor.

Frequently Asked Questions