%5E3-expanded.jpeg "(x+3)^3 Expanded")
Expanding (x + 3)³
The expression (x + 3)³ represents the product of (x + 3) multiplied by itself three times:
(x + 3)³ = (x + 3)(x + 3)(x + 3)
There are two main ways to expand this expression:
1. Using the distributive property (FOIL method)
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Step 1: Expand the first two factors: (x + 3)(x + 3) = x² + 3x + 3x + 9 = x² + 6x + 9
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Step 2: Now multiply the result from Step 1 by (x + 3): (x² + 6x + 9)(x + 3) = x²(x + 3) + 6x(x + 3) + 9(x + 3)
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Step 3: Distribute and simplify: x³ + 3x² + 6x² + 18x + 9x + 27 = x³ + 9x² + 27x + 27
2. Using the Binomial Theorem
The Binomial Theorem provides a formula for expanding expressions of the form (a + b)ⁿ:
(a + b)ⁿ = ∑(n choose k) a^(n-k) b^k
where:
- ∑ represents the sum from k = 0 to n
- (n choose k) is the binomial coefficient, calculated as n! / (k! * (n-k)!)
Applying this to our problem:
- a = x, b = 3, n = 3
Therefore:
(x + 3)³ = (3 choose 0) x³ 3⁰ + (3 choose 1) x² 3¹ + (3 choose 2) x¹ 3² + (3 choose 3) x⁰ 3³
Calculating the binomial coefficients:
- (3 choose 0) = 1
- (3 choose 1) = 3
- (3 choose 2) = 3
- (3 choose 3) = 1
Substituting these values back into the equation:
**(x + 3)³ = x³ + 3x² 3 + 3x 9 + 27 = x³ + 9x² + 27x + 27
Conclusion
Both methods lead to the same expanded form of (x + 3)³ which is x³ + 9x² + 27x + 27. Choose the method that you find easiest to apply.