Step-by-Step Solutions to Polynomials Problems for Class 9
Polynomials are a form of algebraic expressions that consist of variables, coefficients, and constants. This chapter deals with a number of sums focused on simplifying different exponential polynomial expressions. The Polynomials Class 9 worksheet with answers PDF will help you evaluate your understanding of the concepts of this chapter. The Class 9 polynomial worksheet is one of the most useful study resources that aims to teach students the application of various theories of polynomials. Students can self-assess the understanding of the basic concepts by referring to the Worksheet for Class 9 CBSE Maths Polynomials.
About Polynomials
The Polynomials Class 9 worksheet PDF focuses on explaining the term, according to the worksheet, Polynomials are expressions that can be related to one or more terms and used seamlessly with a non-zero coefficient, in a way that it can carry more than one term.
In the polynomials worksheet class 9, each expression that is used in the sum of a polynomial is defined as a term. Let’s suppose that x2 + 5x + 2 is polynomial. In the given example, we can say that the expressions are laid in a way that x2, 5x, and 2 are the terms that are laid in the form of a polynomial. Remember, every single term that is given in a polynomial comprises a coefficient.
Further, the real numbers that are used in the polynomials can also be used to express different terms in the grade 9 math polynomial worksheets. Similar to how the certain numbers are polynomials without any variables, they are known as constant polynomials.
In theory, the constant polynomial 0 is also known as zero polynomial. Degree of the polynomial is the highest power that is available to the suggested polynomial. Consider an example where x3 + y3 + 3xy(x + y), the degree of the polynomial is 3. In a situation where the degree of the sum is a zero, the constant polynomial is a non zero.
Apart from these, polynomials can be further categorised into the suggested three types:
Linear Polynomial – of degree one.
Quadratic Polynomial – of degree two.
Cubic Polynomial – of degree three.
Solved Problems
Q1. Define the suggested degree of each polynomial that is listed below.
(i) 5x3 + 4x2 + 7x
(ii) 4 – y2
(iii) 5t – √7
(iv) 3
Solution:
(i) The given polynomial is 5x3 + 4x2 + 7x.
The suggested equation provides us with a situation where 3 is the highest power of the variable x. So, the degree of the polynomial is 3.
(ii) The given polynomial is 4 - y2. 2 becomes the highest power of the suggested variable that is, y = 2. So, the degree of the polynomial is 2.
(iii) In the suggested polynomial of the situation where 5t – √7. The highest power of variable t is 1. So, the degree of the polynomial is 1.
(iv) Since, 3 = 3x° [∵ x°=1] The equation suggests that the degree of the polynomial for the given equation is a 0.
Q2. Verify whether 2 and 0 are zeroes of the polynomial x2 – 2x.
Solution :
Let p(x) = x2 – 2x
Then p(2) = 22
– 4 = 4 – 4 = 0 and p(0) = 0 – 0 = 0
The solution suggests that the sum 0 and 2 are both the zeroes of the polynomial x2 – 2x.
Listed below are the list of observations around the sums:
(i) The resultant sum of a polynomial doesn’t really have to be a 0.
(ii) The term of a zero polynomial might be a 0.
(iii) Polynomials might comprise of more than one zero
FAQs on Class 9 Polynomials Worksheets – Download & Practice
1. How can practising questions from the Class 9 Maths Polynomials worksheet help in the final exam?
Practising questions from a dedicated worksheet for Class 9 Polynomials is crucial for exam success. It helps students to:
- Strengthen their understanding of core concepts like degree, coefficients, and zeroes of a polynomial.
- Master the application of key theorems, especially the Remainder Theorem and Factor Theorem.
- Improve speed and accuracy in factorising quadratic and cubic polynomials.
- Gain exposure to a variety of question formats, including Higher Order Thinking Skills (HOTS) problems, which go beyond standard textbook exercises.
2. What are the most important types of questions to practice from the Polynomials chapter for the Class 9 exam?
For the Class 9 Maths exam, students should focus on practising these important question types from Polynomials:
- Finding the value of a polynomial for a given value of the variable.
- Identifying the zeroes of a polynomial.
- Applying the Remainder Theorem to find the remainder without performing long division.
- Using the Factor Theorem to determine if a linear polynomial is a factor of another polynomial.
- Factorising cubic polynomials using the Factor Theorem and splitting the middle term.
- Expanding and factorising expressions using algebraic identities.
3. Which algebraic identities from the Polynomials chapter are most frequently tested in Class 9 exams?
The following algebraic identities are considered highly important and are frequently tested in exams:
- (x + y + z)² = x² + y² + z² + 2xy + 2yz + 2zx
- (x + y)³ = x³ + y³ + 3xy(x + y) and (x - y)³ = x³ - y³ - 3xy(x - y)
- x³ + y³ = (x + y)(x² - xy + y²) and x³ - y³ = (x - y)(x² + xy + y²)
- x³ + y³ + z³ - 3xyz = (x + y + z)(x² + y² + z² - xy - yz - zx). This identity is particularly important for 3 or 4-mark questions.
4. What is a common mistake students make when applying the Remainder Theorem, and how can it be avoided?
A very common mistake when using the Remainder Theorem is substituting the incorrect value for the variable. For instance, when a polynomial p(x) is divided by (x + a), many students incorrectly substitute x = a. The correct value to substitute is found by equating the divisor to zero, i.e., x + a = 0, which gives x = -a. To avoid this, always set the divisor equal to zero first to find the correct value that needs to be substituted into the polynomial to find the remainder.
5. How are the 'extra questions' in practice worksheets different from NCERT questions for Polynomials?
NCERT questions build a strong conceptual foundation. Extra questions from practice worksheets are important because they:
- Often have a higher difficulty level and include HOTS (Higher Order Thinking Skills) problems.
- Combine multiple concepts in a single question, for example, using an algebraic identity within a Factor Theorem problem.
- Provide more variety and practice, preparing students for any type of question they might encounter in the exam, thus helping them score higher marks.
6. What is the most efficient method to solve questions on factorising cubic polynomials in an exam?
The most efficient method for factorising a cubic polynomial, as per the CBSE syllabus, is using the Factor Theorem. The steps are:
- Use the 'trial and error' method to find the first zero of the polynomial, p(x). Test simple integer values like 1, -1, 2, -2 that are factors of the constant term.
- If p(a) = 0, then (x - a) is a factor of p(x).
- Divide the cubic polynomial p(x) by the factor (x - a) using the long division method.
- The quotient will be a quadratic polynomial.
- Factorise this quadratic quotient further by using the splitting the middle term method or relevant algebraic identities to get the other two factors.
7. Besides direct factorisation, what is a higher-order application of the identity x³ + y³ + z³ - 3xyz?
A key higher-order application of this identity, often seen in competency-based questions, is to prove conditional results. The most important one is proving that if x + y + z = 0, then x³ + y³ + z³ = 3xyz. This is derived directly from the identity because if the (x + y + z) term is zero, the entire right-hand side becomes zero. This result is then used to evaluate the value of complex-looking expressions without actual calculation, which is a common type of HOTS question.
8. What is the general marks weightage for basic questions on finding the degree or zeroes of a polynomial?
Questions on identifying the degree of a polynomial, classifying polynomials (linear, quadratic, cubic), or finding a simple zero are fundamental concepts. In the Class 9 exam pattern for the session 2025-26, these concepts are typically tested in the form of 1-mark questions, such as Multiple Choice Questions (MCQs) or Very Short Answer (VSA) type questions. While the weightage is low per question, they are very easy to score and essential for building the foundation for more complex problems.
