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:
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A^2 + 2AB + 2AC can be written as A(A + 2B + 2C).
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Combining the terms with B:
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The term B^2 stands alone as it has no other term with B.
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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?
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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.
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Can I apply the same steps to solve (X+Y+Z)^2?
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Yes, the steps for solving (A+B+C)^2 can be applied to any trinomial square, including (X+Y+Z)^2.
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Are there shortcuts or formulas to expand higher powers of trinomials?
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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.
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How can I check my solution for (A+B+C)^2?
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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.
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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.
