Exercise 2.1 — Types and Degree

Practicing the Basics: Degree and Identification of Polynomials

Exercise 2.1 of Class 9 Polynomials and Factorisation is a foundational practice set that helps students apply the definitions learned in the introduction to actual problems. This exercise is important for CBSE, Telangana, and Andhra Pradesh students because it builds the habit of quickly identifying the degree, type, and validity of a polynomial — skills that are tested repeatedly throughout Class 9 and Class 10.

The exercise is organised around six core skills: finding the degree of a polynomial, checking whether an expression is a polynomial in one variable, identifying coefficients of a specific term, classifying polynomials by degree, evaluating true or false statements, and constructing examples of polynomials with a given degree and number of terms.

Finding the Degree of a Polynomial

The first set of problems asks students to find the degree of various polynomials by identifying the term with the highest exponent. Recall that the degree of a polynomial is the largest sum of exponents found in any single term.

A useful tip is to scan every term in the polynomial first, calculate the exponent sum for each, and then pick the largest value as the overall degree. For a constant term like 5, the degree is always 0, since any constant can be expressed as 5x⁰.

Identifying Polynomials in One Variable

This section tests whether students can distinguish between polynomials in one variable, polynomials in two or more variables, and expressions that are not polynomials at all. The reasoning falls into three categories.

• Polynomial in one variable — expressions like 3x² − 2x + 5 or x² + 2 contain only a single variable, x, with whole-number exponents.
• Not a polynomial in one variable — expressions such as p² − 3p + q or x¹⁰⁰ + y¹⁰⁰ involve two different variables, so they cannot be classified as one-variable polynomials.
• Not a polynomial at all — expressions like y + 2/y have a negative exponent, and expressions like 5√x + x√5 contain a variable with a fractional exponent, both of which violate the polynomial condition.

Finding Coefficients and Classifying by Degree

The next part of the exercise focuses on identifying the coefficient of a specific term, usually x³, in a given polynomial. The coefficient is simply the number multiplying that power of the variable. If the term does not appear at all in the polynomial — such as in 2x² + 5 — its coefficient is taken as 0.

Students are also asked to classify polynomials as linear, quadratic, or cubic based on their degree.

• Linear polynomial (degree 1) — examples include x − 1 and 3p.
• Quadratic polynomial (degree 2) — examples include 5x² + x − 7 and πr².
• Cubic polynomial (degree 3) — an example is x − x³.

Recognising these categories quickly is especially useful when moving on to chapters on quadratic equations, where the degree of an expression determines the solution method.

True or False: Testing Conceptual Understanding

The final theoretical section presents statements about polynomials that students must judge as true or false, along with justification. These statements reinforce subtle but important ideas.

• A binomial always has exactly two terms — this is true by definition, regardless of the degree of those terms.
• Not every polynomial is a binomial, since polynomials can have one, two, three, or more terms.
• The degree of the zero polynomial is undefined, not zero — a common point of confusion for students.
• An expression like πr² is a monomial because it has only one term, even though it involves the constant π.

The exercise also asks students to construct their own examples — such as a monomial of degree 10 (like 9x²y⁸) or a trinomial of degree 10 (like 5 − x³ − 8x¹⁰) — which strengthens understanding by working in reverse from the definition to the example.

What This Lesson Prepares You For

Mastering this exercise builds the vocabulary and classification skills needed for the rest of the Polynomials and Factorisation chapter, including finding zeroes of polynomials and applying factorisation identities. These ideas carry forward directly into Class 10, where Exercise 3.3 on zeroes and coefficients and Exercise 3.4 on the division algorithm assume a confident grasp of degree, terms, and classification. Students should revisit the introduction to polynomials if any concept here feels unclear before attempting more advanced exercises.