%5E3-simplified.jpeg "(x-9)^3 Simplified")
Simplifying (x - 9)³
Expanding the expression (x - 9)³ involves multiplying the entire expression by itself three times. This can be done using the distributive property or by using the binomial theorem.
Method 1: Distributive Property
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Square the expression: (x - 9)² = (x - 9)(x - 9) = x² - 18x + 81
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Multiply the result by (x - 9): (x - 9)³ = (x² - 18x + 81)(x - 9) = x²(x - 9) - 18x(x - 9) + 81(x - 9) = x³ - 9x² - 18x² + 162x + 81x - 729
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Combine like terms: (x - 9)³ = x³ - 27x² + 243x - 729
Method 2: Binomial Theorem
The binomial theorem provides a formula for expanding expressions of the form (a + b)ⁿ:
(a + b)ⁿ = ∑_(k=0)^n (n choose k) * a^(n-k) * b^k
where (n choose k) is the binomial coefficient, calculated as n!/(k!(n-k)!).
Applying this to (x - 9)³:
(x - 9)³ = (3 choose 0) * x³ * (-9)⁰ + (3 choose 1) * x² * (-9)¹ + (3 choose 2) * x¹ * (-9)² + (3 choose 3) * x⁰ * (-9)³
= 1 * x³ * 1 + 3 * x² * (-9) + 3 * x * 81 + 1 * 1 * (-729)
= x³ - 27x² + 243x - 729
Therefore, the simplified form of (x - 9)³ is x³ - 27x² + 243x - 729.