Algebraic Identities Practice Questions for CAT 2026 — Formulas and Solved Examples

The Significance of Algebraic Identities

Tackling the Common Admission Test (CAT) can be a daunting task. One essential concept that frequently appears in the exam is algebraic identities. Understanding how to apply these identities is crucial for achieving a high score in this section.

In this blog, we will explore various algebraic identities. These identities will challenge your problem-solving skills. They will also enhance your comprehension of this vital topic. Whether you are preparing for the test for the first time or looking to improve your score, these practice questions will provide you with the necessary tools for success.

Practice with Algebraic Identities

Let's dive into the world of algebraic identities and elevate your CAT exam preparation!

What are Algebraic Identities?

Algebraic identities are equations that hold true for all values of the variables involved. They are fundamental in simplifying expressions and solving equations. Familiarity with these identities can significantly enhance your problem-solving speed and accuracy.

Common Algebraic Identities

Here are some common algebraic identities you should know:

1. Square of a Binomial:
   \[(a + b)^2 = a^2 + 2ab + b^2\]
   \[(a - b)^2 = a^2 - 2ab + b^2\]
   
   
2. Product of a Sum and a Difference:
   \[(a + b)(a - b) = a^2 - b^2\]
   
   
3. Cube of a Binomial:
   \[(a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3\]
   \[(a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3\]
   
   
4. Sum of Cubes:
   \[a^3 + b^3 = (a + b)(a^2 - ab + b^2)\]
   
   
5. Difference of Cubes:
   \[a^3 - b^3 = (a - b)(a^2 + ab + b^2)\]
   
   

Understanding these identities will help you tackle complex problems more efficiently.

Algebraic Identities Questions for CAT

Question Set

Let's put your knowledge to the test with some practice questions.

Answers

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Frequently Asked Questions — Algebraic Identities for CAT 2026

What are algebraic identities and why are they important for CAT?

Algebraic identities are equations that hold true for all values of the variables. In CAT, they appear in algebra, simplification, and polynomial questions. Mastering identities like (a+b)², (a-b)², a³+b³ helps solve problems faster without lengthy calculations.

Which algebraic identities are most tested in CAT 2026?

The most frequently tested identities in CAT are: (a+b)² = a²+2ab+b², (a-b)² = a²-2ab+b², (a+b)(a-b) = a²-b², (a+b)³ = a³+3a²b+3ab²+b³, a³+b³ = (a+b)(a²-ab+b²), and a³-b³ = (a-b)(a²+ab+b²). Vieta's formulas for polynomial roots are also heavily tested.

How many questions on algebra appear in the CAT exam each year?

CAT typically has 3–5 questions from algebra and polynomial-related topics in the QA section. These range from simplification using identities to problems on roots of polynomials, remainder theorem, and max/min of quadratic expressions.

What is the difference between an algebraic identity and an algebraic equation?

An algebraic identity is true for ALL values of the variable (e.g., (a+b)² = a²+2ab+b²), while an algebraic equation is true only for specific values (e.g., x²-5x+6=0 is true only for x=2 and x=3).

How do I use the Remainder Theorem to solve CAT questions?

The Remainder Theorem states that when a polynomial f(x) is divided by (x-a), the remainder is f(a). This is very useful in CAT questions — simply substitute the value of x instead of performing polynomial long division.

What is Vieta's formula and how is it tested in CAT?

Vieta's formulas relate the coefficients of a polynomial to sums and products of its roots. For ax²+bx+c=0: sum of roots = -b/a, product = c/a. For cubic ax³+bx²+cx+d=0: sum of roots = -b/a, sum of products in pairs = c/a, product = -d/a. CAT frequently tests these.

How long does it take to prepare algebraic identities for CAT?

A focused student can master all key algebraic identities in 2–3 days. Spend Day 1 learning and memorising the 10 core identities, Day 2 practising 15–20 application problems, and Day 3 solving timed questions. Regular revision ensures retention.

Are algebraic identities also important for IPMAT, CMAT, and CUET?

Yes. IPMAT (IIM Indore and Rohtak), CMAT, and CUET all test algebraic identities. IPMAT is especially rigorous, often testing polynomial roots and remainder theorem. For all these exams, the same 10–12 core identities are sufficient.

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Frequently Asked Questions — Algebraic Identities for CAT 2026

What are the most important algebraic identities for CAT?

Key identities: (a+b)² = a²+2ab+b², (a-b)² = a²-2ab+b², a²-b² = (a+b)(a-b), (a+b)³ = a³+3a²b+3ab²+b³, (a-b)³ = a³-3a²b+3ab²-b³, a³+b³ = (a+b)(a²-ab+b²), a³-b³ = (a-b)(a²+ab+b²). These are non-negotiable for CAT.

How is (a+b)² used to compute squares quickly?

(a+b)² = a² + 2ab + b². To compute 103²: (100+3)² = 10000 + 600 + 9 = 10609. Similarly 97² = (100-3)² = 10000 - 600 + 9 = 9409. This technique makes squaring 2-3 digit numbers faster than direct multiplication.

What is the a³+b³ identity and when is it useful?

a³+b³ = (a+b)(a²-ab+b²). It is useful when you can identify a sum of cubes pattern. If a+b = 5 and ab = 6, then a³+b³ = (a+b)[(a+b)²-3ab] = 5[25-18] = 35. CAT uses this in simplification and algebraic expression questions.

How do I use identities to simplify expressions with given sums and products?

If x + y = S and xy = P, then: x² + y² = S² - 2P, x³ + y³ = S(S²-3P), x² + y² + z² = (x+y+z)² - 2(xy+yz+zx), x³+y³+z³ - 3xyz = (x+y+z)(x²+y²+z²-xy-yz-zx). These are high-frequency CAT patterns.

What is the Sophie Germain identity?

a⁴ + 4b⁴ = (a²+2b²+2ab)(a²+2b²-2ab). While rarely tested directly, it appears in advanced CAT number theory questions. Knowing it can save time when you see fourth-degree expressions.

How do algebraic identities help in factorisation?

Recognising the pattern a²-b² = (a+b)(a-b) immediately factors expressions like 4x²-9 = (2x+3)(2x-3). Similarly, x³-8 = x³-2³ = (x-2)(x²+2x+4). Factorisation speed directly impacts accuracy in CAT algebra.

What are telescoping products and how are they related to identities?

Telescoping uses (a-b)(a+b) = a²-b² repeatedly. Example: (1-1/4)(1-1/9)(1-1/16)... = (3/4)(8/9)(15/16)... Each factor = (n²-1)/n² = (n-1)(n+1)/n². The product telescopes to a clean answer. CAT tests this in sequence and product problems.

How many algebra questions appear in CAT?

Algebra (including identities, equations, and inequalities) accounts for 5-8 questions in CAT QA section. It is one of the highest-weighted topics. Mastery of identities reduces calculation time and increases accuracy in these questions.