**Introduction**

In algebra, one of the fundamental concepts is understanding and solving equations involving exponents. One such common expression is **(A+B+C)^2**. This may seem daunting at first glance, but with a few simple steps, you can easily expand and simplify this expression to its simplest form. In this guide, we will walk you through the process of solving **(A+B+C)^2** step by step.

**Expanding (A+B+C)^2**

To expand the square of a trinomial such as **(A+B+C)^2**, we will use the distributive property and the rule of exponents. The general formula for expanding a binomial squared is:

**(A+B)^2 = A^2 + 2AB + B^2**

We can extend this formula to a trinomial square by treating **(A+B+C)** as a single term. Therefore, the square of **(A+B+C)** can be expanded as follows:

**(A+B+C)^2 = (A+B+C)(A+B+C)**

Using the distributive property, we can expand this expression further:

= **A(A+B+C) + B(A+B+C) + C(A+B+C)**

= **A^2 + AB + AC + AB + B^2 + BC + AC + BC + C^2**

= **A^2 + 2AB + 2AC + B^2 + 2BC + C^2**

Therefore, **(A+B+C)^2** expands to **A^2 + 2AB + 2AC + B^2 + 2BC + C^2**.

**Simplifying (A+B+C)^2**

After expanding the expression, you can further simplify it by combining like terms. In our expanded form **A^2 + 2AB + 2AC + B^2 + 2BC + C^2**, we can combine the terms that contain the same variables raised to the same powers.

- Combining the terms with
**A**: -
**A^2 + 2AB + 2AC**can be written as**A(A + 2B + 2C)**. -
Combining the terms with

**B**: -
The term

**B^2**stands alone as it has no other term with**B**. -
Combining the terms with

**C**: - Lastly, the term
**C^2**stands alone similar to**B^2**as there are no other terms with**C**.

Therefore, the simplified form of **(A+B+C)^2** is **A(A + 2B + 2C) + B^2 + C^2**.

In conclusion, by following the steps of expanding and simplifying **(A+B+C)^2**, you can easily manipulate trinomial squares and apply these skills to various algebraic problems.

**Frequently Asked Questions (FAQs)**

**What is the difference between expanding and simplifying (A+B+C)^2?**-
**Expanding**involves multiplying out the terms within the squared expression, while**simplifying**focuses on combining like terms to reduce the expression to its simplest form. -
**Can I apply the same steps to solve (X+Y+Z)^2?** -
Yes, the steps for solving

**(A+B+C)^2**can be applied to any trinomial square, including**(X+Y+Z)^2**. -
**Are there shortcuts or formulas to expand higher powers of trinomials?** -
Yes, there are formulas for expanding higher powers such as

**(A+B+C)^3**,**(A+B+C)^4**, and so on, but they involve more complex patterns and terms. -
**How can I check my solution for (A+B+C)^2?** -
You can verify your solution by expanding

**(A+B+C)(A+B+C)**and simplifying the expression to ensure it matches the derived formula**A^2 + 2AB + 2AC + B^2 + 2BC + C^2**. -
**What real-world applications involve (A+B+C)^2 calculations?** - Algebraic expressions like
**(A+B+C)^2**are commonly used in physics, engineering, finance, and computer programming to model and solve various problems and equations.

In mastering the expansion and simplification of **(A+B+C)^2** and similar expressions, you are building a strong foundation in algebra that can be applied to diverse mathematical scenarios.