What Is the Cube of a Binomial Formula with Proof and Solved Examples
Understanding the Cube Of A Binomial is crucial in school algebra, competitive exams, and higher maths. This identity simplifies the expansion of expressions like (a + b)3, which often appear in question papers and real-life calculations. Mastering such patterns gives confidence in solving complex polynomial problems.
Formula Used in Cube Of A Binomial
The standard formula is: \( (a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3 \) and, for subtraction, \( (a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3 \).
Here’s a helpful table to understand Cube Of A Binomial more clearly:
Cube Of A Binomial Table
| Binomial | Expansion | Pattern Visible? |
|---|---|---|
| (x + 2y) | \( x^3 + 6x^2y + 12xy^2 + 8y^3 \) | Yes |
| (a - 3b) | \( a^3 - 9a^2b + 27ab^2 - 27b^3 \) | Yes |
This table shows how the pattern of Cube Of A Binomial appears regularly in algebraic cases.
Worked Example – Solving a Problem
Let’s expand and simplify \( (2x - y)^3 \) step by step using the formula.
1. Write the formula:2. Substitute \( a = 2x \) and \( b = y \):
\( (2x)^3 - 3 \times (2x)^2 \times y + 3 \times (2x) \times y^2 - y^3 \)
3. Calculate powers and simplify coefficients:
\( = 8x^3 - 12x^2y + 6xy^2 - y^3 \)
4. Final Answer:
For further identities and algebraic patterns, see our guide on Algebraic Identities.
Practice Problems
- Expand \( (x + 3y)^3 \) using the cube of a binomial formula.
- Find the expansion of \( (a - 2b)^3 \).
- If \( p + q = 5 \) and \( pq = 6 \), evaluate \( p^3 + q^3 \).
- Compare the expansion of \( (m + n)^3 \) and \( (m - n)^3 \).
You can get more polynomial practice at our Polynomials section.
Common Mistakes to Avoid
- Confusing cube of a binomial with the cube of a single term.
- Missing or misplacing the sign in the expansion, especially with subtraction.
- Incorrectly calculating coefficients (especially 3ab vs 3a2b, etc).
- Forgetting to apply the cube formula—do not treat it as simple multiplication.
Brush up on polynomial expansion techniques here: Expansion Method of Multiplication.
Real-World Applications
The concept of Cube Of A Binomial is used in figuring out volumes, solving higher order equations, and even in advanced topics like the Binomial Theorem. Vedantu explains these skills for classroom and competitive exam success.
We explored the idea of Cube Of A Binomial, learnt expansions with detailed steps, and saw its relevance in bigger problems. Practise regularly and visit Vedantu’s resources to master such algebraic identities for exams and real-life scenarios.
FAQs on Cube of a Binomial Complete Guide with Formula and Expansion
1. What is the formula for the cube of a binomial?
The formula for the cube of a binomial is (a + b)3 = a3 + 3a2b + 3ab2 + b3 and (a − b)3 = a3 − 3a2b + 3ab2 − b3. These formulas come from expanding (a + b)(a + b)(a + b). The coefficients 1, 3, 3, 1 follow the binomial theorem and Pascal’s triangle.
2. How do you expand (a + b)3 step by step?
To expand (a + b)3, multiply the binomial three times and simplify like terms.
- Write: (a + b)(a + b)(a + b)
- First expand (a + b)(a + b) = a2 + 2ab + b2
- Multiply by (a + b): (a2 + 2ab + b2)(a + b)
- Simplify to get: a3 + 3a2b + 3ab2 + b3
3. What is the cube of (a − b)?
The cube of (a − b) is a3 − 3a2b + 3ab2 − b3. Notice the signs alternate because of the negative term. The first and last terms keep their original signs, while the middle terms change according to subtraction.
4. Can you give an example of expanding a cube of a binomial?
Yes, for example, (2x + 3)3 expands using the cube formula.
- Use (a + b)3 = a3 + 3a2b + 3ab2 + b3
- Here, a = 2x and b = 3
- (2x)3 = 8x3
- 3(2x)2(3) = 36x2
- 3(2x)(3)2 = 54x
- 33 = 27
5. Why are the coefficients 1, 3, 3, 1 used in the cube of a binomial?
The coefficients 1, 3, 3, 1 come from the binomial theorem and Pascal’s triangle. In Pascal’s triangle, the fourth row is 1, 3, 3, 1, which represents the coefficients of (a + b)3. These numbers show how many times each term appears when multiplying the binomial three times.
6. What is the difference between (a + b)2 and (a + b)3?
The difference is that (a + b)2 has three terms while (a + b)3 has four terms.
- (a + b)2 = a2 + 2ab + b2
- (a + b)3 = a3 + 3a2b + 3ab2 + b3
7. How do you expand (x − 2y)3?
To expand (x − 2y)3, use the formula for (a − b)3.
- a = x, b = 2y
- a3 = x3
- −3a2b = −3x2(2y) = −6x2y
- +3ab2 = 3x(4y2) = 12xy2
- −b3 = −8y3
8. What are common mistakes when expanding the cube of a binomial?
Common mistakes in the cube of a binomial include incorrect signs and missing middle terms.
- Forgetting the coefficients 3 and 3
- Writing only three terms instead of four
- Not alternating signs correctly in (a − b)3
- Making errors in powers like (a2b)
9. How is the cube of a binomial related to the binomial theorem?
The cube of a binomial is a special case of the binomial theorem when n = 3. The binomial theorem states: (a + b)n = Σ [C(n, r)an−rbr]. For n = 3, the combinations C(3, r) produce coefficients 1, 3, 3, 1, giving the standard cube expansion.
10. Where is the cube of a binomial used in mathematics?
The cube of a binomial is used in algebraic identities, polynomial expansion, and simplifying expressions.
- Solving algebra problems
- Factoring cubic expressions
- Calculus expansions
- Engineering and physics modeling
