Identity Verification: (a−b)^3 MCQs Quiz | Class 9
Welcome to this Class IX Mathematics (Code 041) quiz focused on Unit II: Algebra. This session specifically covers the Verification and Application of the identity (a−b)^3 for expansion and factorization. Solve the 10 multiple-choice questions below to test your understanding, check your score immediately, and download the detailed answer key PDF for your revision.
Understanding the Identity (a – b)^3
In Class IX Algebra, the cubic identity for the difference of two terms is a fundamental concept. It is used extensively for expanding algebraic expressions and factorizing polynomials. The standard formula is:
(a – b)^3 = a^3 – b^3 – 3ab(a – b)
This can also be written in its expanded form by distributing the term -3ab:
(a – b)^3 = a^3 – 3a^2b + 3ab^2 – b^3
Key Concepts for Verification
- Cube of First Term: The first term in the expansion is always positive (a^3).
- Alternating Signs: In the expanded form (a^3 – 3a^2b + 3ab^2 – b^3), the signs alternate: positive, negative, positive, negative.
- Coefficients: The coefficients follow the pattern 1, 3, 3, 1 (ignoring signs).
- Degree Sum: In every term of the expansion, the sum of the powers of ‘a’ and ‘b’ equals 3.
Applications of the Identity
| Action | Example | Process |
|---|---|---|
| Expansion | (2x – 3)^3 | Substitute a=2x, b=3 into the formula. Result: 8x^3 – 36x^2 + 54x – 27. |
| Factorization | x^3 – 8 – 6x(x – 2) | Identify forms matching a^3 – b^3 – 3ab(a-b). Here, it is (x – 2)^3. |
| Calculation | 98^3 | Rewrite as (100 – 2)^3 and apply the formula for easier mental math. |
Quick Revision Points
- Do not confuse (a – b)^3 with a^3 – b^3. They are different expressions.
- Remember that (-b)^3 results in a negative value (-b^3).
- When finding the coefficient of a specific term (like x^2), pay close attention to the negative sign in -3a^2b.
Extra Practice Questions
Try solving these additional problems to strengthen your grasp on the topic:
- 1. Expand (3p – 4q)^3 completely.
- 2. Factorize 27a^3 – 125b^3 – 135a^2b + 225ab^2.
- 3. Without direct multiplication, evaluate 998^3.
- 4. If x – 1/x = 3, find the value of x^3 – 1/x^3.
- 5. Verify if (x – 2y)^3 is equal to x^3 – 8y^3 – 6xy(x – 2y).
