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Simplifying (2x + 1)³
Expanding and simplifying algebraic expressions is a fundamental skill in mathematics. One common type of expression involves raising a binomial to a power, such as (2x + 1)³. Let's break down how to simplify this expression.
Understanding the Problem
The expression (2x + 1)³ represents (2x + 1) multiplied by itself three times:
(2x + 1)³ = (2x + 1) * (2x + 1) * (2x + 1)
To simplify this, we can use the distributive property (also known as FOIL) to multiply the terms.
Step-by-Step Solution
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Multiply the first two binomials:
(2x + 1) * (2x + 1) = 4x² + 2x + 2x + 1 = 4x² + 4x + 1
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Multiply the result from step 1 by the remaining binomial:
(4x² + 4x + 1) * (2x + 1) = 8x³ + 4x² + 8x² + 4x + 2x + 1
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Combine like terms:
8x³ + 4x² + 8x² + 4x + 2x + 1 = 8x³ + 12x² + 6x + 1
Final Answer
Therefore, the simplified form of (2x + 1)³ is 8x³ + 12x² + 6x + 1.
Using the Binomial Theorem
For higher powers, the binomial theorem offers a more efficient way to expand binomials. However, for a simple case like (2x + 1)³, the step-by-step method illustrated above is sufficient.