How to Expand (A-B)^3: Formula and Example

Expanding a binomial expression raised to a power is a fundamental concept in algebra. One common expansion problem students encounter is ‘(a – b)^3‘. In this article, we will explore the formula for expanding (a – b)^3 using the binomial theorem and walk through an example to demonstrate the process step by step.

Formula for Expanding (a – b)^3:

The expansion of (a – b)^3 can be calculated using the binomial theorem. The general formula for expanding (a – b)^n is:

[a^n - nC1 * a^(n-1) * b + nC2 * a^(n-2) * b^2 - ... + (-1)^n * b^n]

where nCk represents the binomial coefficient, calculated as n! / k!(n-k)!.

Expanding (a – b)^3:

Let’s apply the formula to expand (a – b)^3:

1. n = 3:
   The terms in the expansion of (a – b)^3 will include a^3, -3a^2b, 3ab^2, and -b^3.

2. Substitute into the formula:

3. a^3
4. – 3(3C1) * a^2b = -3a^2b
5. + 3(3C2) * ab^2 = 3ab^2

6. – b^3

7. Calculate the coefficients:

8. 3C1 = 3

9. 3C2 = 3

10. Combine the terms:
    The expanded form of (a – b)^3 is: a^3 - 3a^2b + 3ab^2 - b^3.

Therefore, (a – b)^3 = a^3 – 3a^2b + 3ab^2 – b^3.

This expansion process can be applied to raise any binomial expression to a power, providing a systematic way to expand and simplify complicated algebraic equations.

Example:

Let’s work through an example to illustrate the expansion of (a – b)^3:

Given: (a - b)^3

1. Apply the formula:

2. a^3 – 3a^2b + 3ab^2 – b^3

3. Substitute a = 2, b = 1:

4. (2)^3 – 3(2)^2(1) + 3(2)(1)^2 – (1)^3
5. 8 – 12 + 6 – 1
6. 1

Therefore, (2 – 1)^3 = 1.

Frequently Asked Questions (FAQs):

1. What is the binomial theorem?

2. The binomial theorem is a mathematical theorem that describes the algebraic expansion of powers of binomials.

3. How do you calculate binomial coefficients?

4. Binomial coefficients are calculated using the formula n! / k!(n-k)!, where n! represents the factorial of n.

5. Why is expanding binomial expressions important?

6. Expanding binomial expressions helps simplify algebraic equations and allows for easier manipulation of mathematical expressions.

7. Can the binomial theorem be applied to any power of a binomial?

8. Yes, the binomial theorem can be applied to any power of a binomial with positive integer exponents.

9. What are some real-life applications of expanding binomial expressions?

10. Expanding binomial expressions is essential in fields like physics, engineering, and computer science for solving complex problems and modeling real-world phenomena.

In conclusion, understanding how to expand binomial expressions like (a – b)^3 is crucial for mastering algebraic manipulations. By following the formula and working through examples, students can enhance their problem-solving skills and tackle more advanced mathematical concepts with confidence.