Exercise 2.5 — Algebraic Identities

Exercise 2.5 Polynomials and Factorisation – Algebraic Identities Made Simple

Exercise 2.5 from Chapter 2, Polynomials and Factorisation, is one of the most important exercises for Class 9 students following the CBSE, Telangana, and Andhra Pradesh syllabus. This exercise focuses on applying standard algebraic identities to multiply expressions quickly, evaluate large numbers without long multiplication, factorise polynomials, and solve word problems involving area and volume. Mastering these identities builds a strong foundation for later topics like quadratic equations and coordinate geometry.

Key Identities Used in Exercise 2.5

Before solving the problems, students must be confident with the following identities. These appear repeatedly throughout the exercise and form the backbone of every solution.

• (x + y)² = x² + 2xy + y² — used for squaring a sum
• (x − y)² = x² − 2xy + y² — used for squaring a difference
• (x + y)(x − y) = x² − y² — the difference of squares identity, very useful for quick multiplication of numbers close to a round figure
• (x + a)(x + b) = x² + (a + b)x + ab — used to multiply two binomials and to factorise quadratic trinomials
• (x + y + z)² = x² + y² + z² + 2xy + 2yz + 2zx — extends the square identity to three terms
• (x + y)³ = x³ + 3x²y + 3xy² + y³ and (x − y)³ = x³ − 3x²y + 3xy² − y³ — used for cubing binomials
• x³ + y³ + z³ − 3xyz = (x + y + z)(x² + y² + z² − xy − yz − zx) — a powerful identity used in the later, more advanced questions

Multiplying Expressions Using Identities

The first set of questions asks students to find products such as (x + 5)(x + 2), (x − 5)(x − 5), and (3x + 2)(3x − 2) without performing direct multiplication. By recognising the pattern, (x + 5)(x + 2) matches the identity (x + a)(x + b), giving the answer x² + 7x + 10 instantly. Similarly, (3x + 2)(3x − 2) is a direct application of the difference of squares identity, simplifying to 9x² − 4. These questions train students to "see" the identity hidden inside an expression, which is a critical exam skill for both objective and descriptive questions.

Evaluating Numbers Without Direct Multiplication

One of the most exam-favourite parts of this exercise involves evaluating products like 101 × 99, 999 × 999, and 501 × 501 using identities instead of long multiplication. For example, 101 × 99 is rewritten as (100 + 1)(100 − 1), which equals 100² − 1² = 9999 using the difference of squares identity. Likewise, 999 × 999 becomes (1000 − 1)², expanded using (x − y)² to give 998001. This technique saves valuable time during board exams and is especially useful for mental maths and competitive exam preparation.

Factorisation Using Standard Identities

The next major skill covered is factorisation. Expressions like 16x² + 24xy + 9y² and 4y² − 4y + 1 are recognised as perfect square trinomials and factorised as (4x + 3y)² and (2y − 1)² respectively. Other expressions such as 18a² − 50 use the difference of squares pattern after taking out a common factor, giving 2(3a + 5)(3a − 5). Trinomials like x² + 5x + 6 and 3p² − 24p + 36 are factorised by splitting the middle term using the identity x² + (a + b)x + ab = (x + a)(x + b).

Expanding Three-Term and Cubic Expressions

Questions involving three-term squares, such as (x + 2y + 4z)² and (−2a + 5b − 3c)², apply the extended identity x² + y² + z² + 2xy + 2yz + 2zx. Cubic expansions like (2a − 3b)³ and (p + 1)³ apply the cube identities for sums and differences of two terms. These problems also reinforce factorisation in reverse — for instance, expressions like 8a³ + b³ + 12a²b + 6ab² can be factorised back into (2a + b)³ using the identity x³ + y³ + 3xy(x + y) = (x + y)³.

Special Values, Cubes, and Word Problems

The later questions build toward more advanced applications. Students verify identities like x³ + y³ = (x + y)(x² − xy + y²) using actual numbers, then use this to factorise expressions such as 27a³ + 64b³ and 343y³ − 1000. The exercise also introduces a very important special case: when x + y + z = 0, then x³ + y³ + z³ = 3xyz. This result is used to evaluate expressions like (−10)³ + 7³ + 3³ without calculating large cubes directly. Finally, the exercise connects algebra to geometry through problems on finding the possible length and breadth of a rectangle (e.g., 4a² + 4a − 3) and the dimensions of a cuboid (e.g., 3x³ − 12x), helping students see how factorisation applies to real measurement problems.

Tips for Scoring Well in This Exercise

• Always write down the identity being used before substituting values — examiners award marks for showing the correct method
• Practice recognising patterns: look for perfect squares, difference of squares, and sum/difference of cubes before attempting long multiplication
• For numerical evaluation problems, try to express the number as a round number plus or minus a small number
• Remember the special identity for x + y + z = 0, as it appears frequently in CBSE, Telangana, and Andhra Pradesh board exams
• Practice factorisation in both directions — expanding an identity and recognising an expanded form to factorise it

What This Lesson Prepares You For

A strong grip on Exercise 2.5 prepares students for more advanced algebra topics. To strengthen related concepts, students can revisit the basics of introduction to polynomials and practice Exercise 2.4 on factorisation. These identities also form the foundation for quadratic equations in Class 10, where factorisation skills are extended to solve equations. Students preparing for board exams across CBSE, Telangana, and Andhra Pradesh boards should treat this exercise as a core revision topic before moving to higher-order algebra chapters.