When it comes to expanding and simplifying expressions like **(A-B)^3**, it is essential to utilize the binomial theorem. This theorem provides a neat way to raise a binomial (an expression with two terms) to any power, including cubing it.

The formula for expanding **(A-B)^3** is as follows:

**(A-B)^3 = A^3 - 3A^2B + 3AB^2 - B^3**.

Let's break down the process to get a better understanding of how this expansion works:

### Understanding the Expansion Process:

#### Step 1: Cube the binomial

Start by cubing **(A-B)**, which can be done by multiplying it by itself three times:

**(A-B)(A-B)(A-B)**.

#### Step 2: Apply the FOIL method

Now, apply the FOIL method (First, Outer, Inner, Last) to each pair of terms in the brackets:

- First:
**A**.*A*A = A^3 - Outer:
**A**.*(-B)*A = -A^2B - Inner:
**-B**.*A*A = -AB^2 - Last:
**(-B)**.*(-B)*(-B) = -B^3

#### Step 3: Combine the results

Combine the results from each step to get the final expanded form:

**(A-B)^3 = A^3 - 3A^2B + 3AB^2 - B^3**.

### Simplifying the Expansion:

Once you have expanded **(A-B)^3** using the formula above, you can simplify the expression by combining like terms. Make sure to pay attention to the coefficients of the terms and keep the variables in line:

**A^3 - 3A^2B + 3AB^2 - B^3**.

And there you have it - the expanded and simplified form of **(A-B)^3** using the binomial theorem. This process can be a useful tool when dealing with higher powers of binomials in algebraic expressions.

### Frequently Asked Questions (FAQs):

#### Q1: What is the binomial theorem?

A1: The binomial theorem is a mathematical formula that describes the algebraic expansion of powers of binomials.

#### Q2: Why is the FOIL method important in expanding binomials?

A2: The FOIL method (First, Outer, Inner, Last) is essential in expanding binomials as it provides a systematic way to multiply the terms of two binomials.

#### Q3: Can the binomial theorem be applied to binomials raised to powers other than three?

A3: Yes, the binomial theorem can be applied to expand binomials raised to any positive integer power.

#### Q4: How do you simplify expressions after expanding them using the binomial theorem?

A4: Simplify expressions by combining like terms, ensuring that the coefficients of the terms are appropriately managed.

#### Q5: Are there any shortcuts for expanding binomials to higher powers?

A5: Some advanced techniques, such as the Pascal's triangle or combinatorial methods, can be used to quickly expand binomials to higher powers, especially for larger exponents.

By following these steps and understanding the principles behind the binomial theorem, you can confidently expand and simplify expressions like **(A-B)^3** in algebraic calculations.