%5E3-simplify.jpeg "(x-4)^3 Simplify")
Simplifying (x - 4)^3
The expression (x - 4)^3 represents the product of (x - 4) multiplied by itself three times. To simplify this expression, we can expand it using the distributive property or by using the binomial theorem.
Expanding using the distributive property:
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Expand the first two factors: (x - 4)(x - 4) = x(x - 4) - 4(x - 4) = x² - 4x - 4x + 16 = x² - 8x + 16
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Multiply the result by (x - 4): (x² - 8x + 16)(x - 4) = x(x² - 8x + 16) - 4(x² - 8x + 16) = x³ - 8x² + 16x - 4x² + 32x - 64
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Combine like terms: x³ - 8x² - 4x² + 16x + 32x - 64 = x³ - 12x² + 48x - 64
Therefore, the simplified form of (x - 4)³ is x³ - 12x² + 48x - 64.
Using the binomial theorem:
The binomial theorem provides a formula to expand expressions of the form (a + b)ⁿ. In this case, a = x, b = -4, and n = 3.
The binomial theorem states: (a + b)ⁿ = ∑ (n choose k) a^(n-k) b^k
where (n choose k) represents the binomial coefficient, calculated as n! / (k! * (n-k)!).
Applying this to our expression:
(x - 4)³ = (3 choose 0) x³ (-4)⁰ + (3 choose 1) x² (-4)¹ + (3 choose 2) x¹ (-4)² + (3 choose 3) x⁰ (-4)³
Calculating the binomial coefficients:
- (3 choose 0) = 1
- (3 choose 1) = 3
- (3 choose 2) = 3
- (3 choose 3) = 1
Substituting these values and simplifying:
(x - 4)³ = 1 * x³ * 1 + 3 * x² * (-4) + 3 * x * 16 + 1 * 1 * (-64) = x³ - 12x² + 48x - 64
As you can see, both methods lead to the same simplified form: x³ - 12x² + 48x - 64.
The choice of method depends on your preference and the complexity of the expression. For simple cases like this, the distributive property might be more intuitive. However, for higher powers or more complex expressions, the binomial theorem offers a more structured and efficient approach.