%5E3-expand.jpeg "(x-7)^3 Expand")
Expanding (x-7)³
Expanding a binomial raised to a power can be done using the binomial theorem or by repeated multiplication. Let's explore both methods for expanding (x-7)³.
Using the Binomial Theorem
The binomial theorem states:
(a + b)ⁿ = ∑(n choose k) a^(n-k) b^k
where:
- n is the power
- k is a value from 0 to n
- (n choose k) is the binomial coefficient, calculated as n! / (k! * (n-k)!)
Applying this to our problem:
- a = x
- b = -7
- n = 3
Let's expand it step-by-step:
(x - 7)³ = ∑(3 choose k) x^(3-k) (-7)^k
- k = 0: (3 choose 0) x³ (-7)⁰ = 1 * x³ * 1 = x³
- k = 1: (3 choose 1) x² (-7)¹ = 3 * x² * (-7) = -21x²
- k = 2: (3 choose 2) x¹ (-7)² = 3 * x * 49 = 147x
- k = 3: (3 choose 3) x⁰ (-7)³ = 1 * 1 * (-343) = -343
Therefore, the expanded form of (x-7)³ is:
(x - 7)³ = x³ - 21x² + 147x - 343
Using Repeated Multiplication
We can also expand (x-7)³ by multiplying (x-7) by itself three times.
- (x - 7) * (x - 7) = x² - 14x + 49 (using the FOIL method)
- (x² - 14x + 49) * (x - 7) = x³ - 21x² + 147x - 343 (using distributive property)
As you can see, we arrive at the same expanded form using both methods.
Conclusion
Both the binomial theorem and repeated multiplication provide valid ways to expand (x-7)³. The binomial theorem offers a more concise and systematic approach, especially for higher powers, while repeated multiplication can be more intuitive for smaller powers. Choosing the method depends on your personal preference and the complexity of the problem.