CBSE Notes Class 9 Maths Chapter 2 Polynomial
Polynomials are algebraic expressions made using variables, constants, and powers, and they play a major role in solving mathematical equations. They help us describe patterns, relationships, and real-life quantities in a simple symbolic form, making algebra easier to understand and apply.
In CBSE Maths Chapter 2 Polynomials Class 9 Notes, the chapter introduces types of polynomials, degree of a polynomial, zeroes of a polynomial, and the relationship between zeroes and coefficients of linear and quadratic polynomials. These CBSE Class 9 Maths notes explain each concept in a clear and step-wise manner, along with easy examples that strengthen conceptual understanding and support accurate problem solving in exams.
1.0Download CBSE Class 9 Maths Notes Chapter 2 Polynomial - Free PDF
Master the concepts of Polynomials with ease! Our free CBSE Class 9 Maths Chapter 2 Notes PDF are carefully prepared to help students grasp the fundamentals quickly and effectively.
2.0CBSE Class 9 Maths Notes Chapter 2 Polynomial - Revision Notes
Important Concepts in Polynomials
Polynomial Definition: A polynomial is an algebraic expression consisting of variables, coefficients, and non-negative integer exponents.
Types of Polynomials:
- Monomial: A polynomial with one term (e.g., 5x).
- Binomial: A polynomial with two terms (e.g., x2+ 3).
- Trinomial: A polynomial with three terms (e.g., x3 + x2+ x ).
Degree and Zeros of a Polynomial
- Degree of a Polynomial: The degree of a polynomial is often defined as the highest power of any variable in the expression. For example, in 4x3 + 2x + 1, the degree is 3.
- Zeros of a Polynomial: The zeros of any polynomial are the variable values that make the polynomial equal to zero. For example, if p (x) = xβ3, then x = 3 is a zero of the polynomial.
Definitions
- Constant Polynomial: A polynomial with degree 0 (e.g., 5).
- Linear Polynomial: A polynomial of degree 1 (e.g., 3x + 4).
- Quadratic Polynomial: A polynomial of degree 2 (e.g., x2 β 5x + 6).
- Cubic Polynomial: A polynomial of degree 3 (e.g., x3+ 2x2β x).
- Remainder Theorem: States that if a polynomial p (x) is divided by xβa the remainder is p(a).
- Factor Theorem: If p(a) = 0, then xβa is a factor of p (x).
Formulas
- Remainder Theorem: Remainder =p(a) when p(x) is divided by xβa
- Factor Theorem: p(a)=0βΉ(xβa) is a factor of p(x)
- Polynomials in Division:
Dividend = (Divisor Γ Quotient) + Remainder
Or,
p (x) = (xβa) q (x)+ r
Here, q (x) is the quotient, and r is the remainder.
For the division of polynomials, follow the following steps
Step 1: Write the terms of both polynomials in descending order of degrees (if they are not already in that order).
Step 2: Divide the first term of the dividend (the polynomial you are dividing) by the first term of the divisor (the polynomial you are dividing by).
Step 3: Multiply the whole divisor by the term you just found (from step 2).
Step 4: Subtract the result obtained in Step 3 from the dividend.
Step 5: Now repeat the division with a new polynomial.
Step 6: Subtract again
Step 7: Check whether the degree of the remainder is less than the degree of the divisor.
Tips and Tricks
- Identifying the Degree Quickly: Focus on the term with the highest power to determine the degree of the polynomial.
- Applying Remainder Theorem: Substitute x = a directly into the polynomial to quickly find the remainder.
- Factorization Made Easy: Use the Factor Theorem to check for simple factors like xβ1 or x+2.
- Understanding Graphs: Visualise the zeros of polynomials as points where the graph intersects the x-axis.
- A table summarising the types of polynomials and their degrees for quick revision:
3.0Solved Problems
Example 1: Divide the polynomial P(x) = 2x3β3x2+4xβ5 by (xβ2). Find the remainder using the Remainder Theorem.
Solution: give that x-2 is divisor
x - 2 = 0; x = 2
P(x) = 2x3 β 3x2 + 4x β 5 put x = 2
P(2) = 2 x 23β3 x 22+ 4 x 2 β 5
= 16 - 12 + 8 - 5 = 7
Example 2: Determine if (xβ2) is a factor of the polynomial P(x) = x3 β 4x2 + 3x + 2.
Solution: x-2 = 0; x = 2
Put x = 2 in P(x) = x3 β 4x2 + 3x + 2 to check whether (x-2) is factor or not.
P(2) = 23 β 422 + 3 x 2 + 2
P(2) = 8 β 16 + 6 + 2
= 16 β 16 = 0
Since p(2) = 0, hence x-2 is a factor of the given equation.
Example 3: Find the remainder when p(x)=x3β4x+6 is divided by π₯β2.
Solution:
Using the Remainder Theorem, substitute x=2 into p (x):
P(2)=23β4(2)+6=8β8+6=6.
The remainder is 6.
Example 4: Divide 3x4 + x3 - 17x2 + 19x - 6 by 3x2 + 7x - 6.
Solution:
Here, in this example, the remainder is 0.
Example 5: Find the value of βaβ if x β a is a factor of x3 β ax2 + 2x + a β 1.
Solution:
Let p(x) = x3 β ax2 + 2x + a β 1
Since (x β a) is a factor of p(x), p(a) = 0.
x - a = 0; x = a
p(a) = a3 β a(a)2 + 2a + a β 1 = 0
a3 β a3 + 2a + a β 1 = 0
3a = 1
Hence, a = 1/3
Example 6: If x + y = 12 and xy = 27, find the value of x3 + y3.
Solution:
Using identity
= (x+y)2 = x2 + 2xy + y2
= 122 = x2 + y2 + 227
= x2 + y2 = 144 - 54 = 90
Now, x3 + y3 = (x + y) (x2 + y2β xy)
= 12 [90 - 27]
= 12 Γ 63 = 756
Example 7: Factorise: 9x2 + 4y2 + 16z2 + 12xy β 16yz β 24xz
Solution:
Using the identity
(x + y + z)2 = x2 + y2 + z2 + 2xy + 2yz + 2zx
We can also write this equation 9x2 + 4y2 + 16z2 + 12xy β 16yz β 24xz as
=(3x)2+(2y)2+(β4z)2+2Γ3xΓ2yβ2Γ2yΓβ4z+2Γ3xΓβ4z
= (3x+2y-4z)2 = (3x+2y-4z)(3x+2y-4z)
4.0Key Features of CBSE Class 9 Maths Chapter 2 Polynomial Notes
- Concept-Focused Explanation β The notes clearly explain the definition of a polynomial, terms, coefficients, variables, and degree with simple language so students can build a strong algebraic foundation.
- Types of Polynomials β Covers linear, quadratic, cubic, and higher-degree polynomials along with classification based on number of terms (monomial, binomial, trinomial) for better conceptual clarity.
- Standard Form & Identification β Helps students identify standard form, leading coefficient, constant term, and degree β important for solving NCERT and exam-oriented questions.
- Solved Examples (NCERT Based) β Important textbook problems solved in a step-wise method to match CBSE exam pattern.
- CBSE Pattern Coverage β Prepared strictly according to the latest CBSE syllabus to help students score better in school exams and competitive exams.
