How to Derive and Use the a³ – b³ Formula with Solved Examples
The concept of a cube minus b cube formula is essential in mathematics and helps in solving real-world and exam-level problems efficiently.
Understanding A Cube Minus B Cube Formula
The a cube minus b cube formula is an algebraic identity used to factorise expressions of the form \( a^3 - b^3 \). It explains how to break down the difference of cubes into a product of two factors. This concept is widely used in polynomial factorization, simplifying algebraic expressions, and solving equations, especially in class 9 and class 10 maths.
Formula Used in A Cube Minus B Cube Formula
The standard formula is: \( a^3 - b^3 = (a - b)(a^2 + ab + b^2) \)
Here’s how the a cube minus b cube formula works step-by-step:
1. Identify expressions in the cube form, such as \( a^3 - b^3 \).
2. Write the formula: \( a^3 - b^3 = (a - b)(a^2 + ab + b^2) \).
3. Substitute your values for 'a' and 'b' to factorise or simplify.
Stepwise Proof of A Cube Minus B Cube Formula
Let’s prove the formula for a cube minus b cube using expansion:
1. Start with the identity: \( (a - b)(a^2 + ab + b^2) \)
2. Expand the multiplication:
3. Multiply terms out:
4. Simplify like terms:
\( + a^2b - a^2b = 0 \), \( + ab^2 - ab^2 = 0 \)
\( - b^3 \)
5. Result:
So, the formula is proven: \( a^3 - b^3 = (a - b)(a^2 + ab + b^2) \)
Difference Between a³ – b³, (a – b)³, and a³ + b³
Students often confuse different cube and binomial formulas. Here’s a comparison:
| Formula | Expanded Form | Usage |
|---|---|---|
| \( a^3 - b^3 \) | \( (a - b)(a^2 + ab + b^2) \) | Difference of cubes |
| \( a^3 + b^3 \) | \( (a + b)(a^2 - ab + b^2) \) | Sum of cubes |
| \( (a - b)^3 \) | \( a^3 - 3a^2b + 3ab^2 - b^3 \) | Cube of a binomial |
Worked Example – Solving a Problem
Let’s solve a problem using the a cube minus b cube formula:
1. Factorise \( 64x^3 - 216 \):
2. Write 64x³ and 216 as cubes:
3. Apply the formula:
4. Substitute:
5. Simplify inside the brackets:
6. Final answer:
Practice Problems
- Factorise \( 125y^3 - 8 \) using the a cube minus b cube formula.
- Solve: If \( a = 5 \), \( b = 2 \), calculate \( a^3 - b^3 \).
- Find the value of \( x \) if \( x^3 - 27 = 0 \).
- Simplify \( 1000 - 343 \) using the difference of cubes identity.
Common Mistakes to Avoid
- Confusing a cube minus b cube formula with (a – b)³ expansion.
- Missing the middle term (ab) in the trinomial for the factorization.
- Switching plus and minus signs between a³ – b³ and a³ + b³ identities.
- Not recognising perfect cubes in expressions before applying the formula.
Real-World Applications
The concept of a cube minus b cube formula appears in advanced calculations like finding volumes, solving higher degree equations, coding, engineering applications, and in competitive exams. Vedantu helps students see how maths applies beyond the classroom, especially by connecting algebraic identities to problem-solving and real-life context.
We explored the idea of a cube minus b cube formula, how to apply it, solve related problems, and understand its real-life relevance. Practice more with Vedantu to build confidence in these concepts.
Related Topics to Explore
- A Cube Plus B Cube Formula – Compare the sum of cubes with the difference.
- Algebraic Identities – Learn all standard identities in school maths.
- Factorization of Algebraic Expressions – Master methods to break expressions into simpler factors.
- Polynomial – Understand polynomials and their factorization techniques.
- Cubes and Cube Roots – Deepen your conceptual understanding of cubes.
FAQs on A Cube Minus B Cube Formula Explained for Students
1. What is the a cube minus b cube formula?
The a cube minus b cube formula is an important algebraic identity used to factor the difference between two cubes. It is expressed as a³ – b³ = (a – b)(a² + ab + b²). This formula simplifies solving polynomial equations and is widely applicable in board exams and competitive tests.
2. How to derive a³ – b³ step by step?
To derive a³ – b³ = (a – b)(a² + ab + b²), follow these steps:
1. Consider RHS: (a – b)(a² + ab + b²)
2. Expand: a(a² + ab + b²) – b(a² + ab + b²)
3. Simplify: a³ + a²b + ab² – a²b – ab² – b³
4. Cancel common terms: a²b – a²b and ab² – ab²
5. Result: a³ – b³ equals LHS
This stepwise proof confirms the identity.
3. Where is a³ – b³ used in class 10 maths?
The a³ – b³ formula is used in many class 10 maths topics including:
• Factorization of polynomials
• Solving cubic equations
• Simplifying algebraic expressions involving cubes
• Competitive exams revision
Understanding this formula helps students quickly solve complex problems involving cubes.
4. What is the difference between (a – b)³ and a³ – b³?
The expressions (a – b)³ and a³ – b³ are different:
• (a – b)³ expands to a³ – 3a²b + 3ab² – b³, involving additional terms.
• a³ – b³ factors as (a – b)(a² + ab + b²) and is used for difference of cubes.
Confusing these leads to errors in factorization and expansion; always remember the distinct formulas.
5. Can you show an example using a³ – b³?
Certainly! Example: Factorize 64x³ – 216
1. Recognize cubes: 64x³ = (4x)³, 216 = 6³
2. Apply formula: a³ – b³ = (a – b)(a² + ab + b²)
3. Substitute: (4x – 6)(16x² + 24x + 36)
This factorization simplifies solving polynomial problems.
6. Why can’t I factor a³ – b³ using (a – b)(a² – ab + b²)?
The factorization of a³ – b³ requires (a – b)(a² + ab + b²). Using (a – b)(a² – ab + b²) is incorrect because the middle term's sign must be positive to correctly reconstruct the difference of cubes. Using the wrong sign results in an expression that does not simplify back to a³ – b³.
7. Why do many students mix up a³ – b³ and (a – b)³ in board exams?
Students often confuse a³ – b³ with (a – b)³ because both involve cubes and subtraction, but their expansions differ. The cube of a binomial (a – b)³ includes additional terms with coefficients 3, while a³ – b³ factors into two polynomials. Clear understanding and memorization of the distinct formulas prevent this common mistake.
8. How do I spot “difference of cubes” in tricky polynomials?
To identify a difference of cubes:
• Check if both terms are perfect cubes (e.g., 27 = 3³, x³ = x³)
• Confirm subtraction between two cubic terms
• If both conditions match, use a³ – b³ formula for factorization
Spotting these helps in solving complex polynomial expressions quickly.
9. Why is a³ – b³ important for JEE or Olympiad?
The a³ – b³ identity is vital in competitive exams like JEE and Math Olympiads because:
• It simplifies complex polynomial expressions
• Saves time during problem solving
• Helps in factoring cubic expressions efficiently
Mastering this formula is crucial for scoring well in algebra-intensive exams.
10. What are common calculation mistakes while expanding a³ – b³?
Common mistakes include:
• Incorrect sign in the binomial or trinomial part
• Confusing a³ – b³ with (a – b)³
• Miscalculating expansion terms and missing middle terms
• Forgetting that the middle term in factorization uses a positive sign
A careful stepwise approach reduces errors.
11. What is the formula for a³ + b³?
The formula for a³ + b³ is a³ + b³ = (a + b)(a² – ab + b²). It factors the sum of cubes and is similarly important in algebra and competitive exam problems.
12. How is the a³ – b³ formula different from (a + b)³?
While a³ – b³ factors into (a – b)(a² + ab + b²), the expansion of (a + b)³ is a³ + 3a²b + 3ab² + b³. The difference lies in factorization versus binomial expansion, each serving distinct purposes in algebraic manipulation.
