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Simplifying the Expression (2x - y)³
This article explores how to simplify the expression (2x - y)³.
Understanding the Concept
The expression (2x - y)³ represents the product of (2x - y) multiplied by itself three times:
(2x - y)³ = (2x - y) * (2x - y) * (2x - y)
Expanding the Expression
To simplify, we need to expand the expression. This can be done using the distributive property (also known as FOIL) or by using the binomial theorem.
Using the Distributive Property:
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First, expand (2x - y) * (2x - y): (2x - y) * (2x - y) = 4x² - 2xy - 2xy + y² = 4x² - 4xy + y²
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Now, multiply the result by (2x - y): (4x² - 4xy + y²) * (2x - y) = 8x³ - 8x²y + 2xy² - 4x²y + 4xy² - y³
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Combine like terms: 8x³ - 12x²y + 6xy² - y³
Using the Binomial Theorem:
The binomial theorem provides a formula for expanding expressions of the form (a + b)ⁿ. Applying this to our expression:
(2x - y)³ = ¹C₀ (2x)³ (-y)⁰ + ¹C₁ (2x)² (-y)¹ + ¹C₂ (2x)¹ (-y)² + ¹C₃ (2x)⁰ (-y)³
Simplifying this gives us:
(2x - y)³ = 8x³ - 12x²y + 6xy² - y³
Final Result
Therefore, the simplified form of (2x - y)³ is 8x³ - 12x²y + 6xy² - y³.