%5E6.jpeg "(x-2)^6")
Expanding (x-2)^6
The expression (x-2)^6 represents the sixth power of the binomial (x-2). Expanding this expression involves applying the binomial theorem, a powerful tool for handling powers of binomials.
Understanding the Binomial Theorem
The binomial theorem states that for any real numbers x and y and any non-negative integer n:
(x + y)^n = Σ (n choose k) x^(n-k) y^k
where the summation runs from k = 0 to n, and (n choose k) represents the binomial coefficient, calculated as:
(n choose k) = n! / (k! * (n-k)!)
Applying the Binomial Theorem to (x-2)^6
In our case, x = x, y = -2, and n = 6. We can use the binomial theorem to expand (x-2)^6 as follows:
(x - 2)^6 = (6 choose 0) x^6 (-2)^0 + (6 choose 1) x^5 (-2)^1 + (6 choose 2) x^4 (-2)^2 + (6 choose 3) x^3 (-2)^3 + (6 choose 4) x^2 (-2)^4 + (6 choose 5) x^1 (-2)^5 + (6 choose 6) x^0 (-2)^6
Now we calculate the binomial coefficients and simplify:
(x - 2)^6 = 1x^6 - 12x^5 + 60x^4 - 160x^3 + 240x^2 - 192x + 64
Final Result
Therefore, the expanded form of (x-2)^6 is x^6 - 12x^5 + 60x^4 - 160x^3 + 240x^2 - 192x + 64.
Importance of Understanding Binomial Expansion
Understanding the binomial theorem and its applications is crucial in various fields, including:
- Algebra: Simplifying complex expressions involving powers of binomials.
- Calculus: Deriving formulas for derivatives and integrals of functions involving powers of binomials.
- Probability: Calculating probabilities related to binomial distributions.
- Statistics: Analyzing data and making inferences using binomial models.
The ability to expand expressions like (x-2)^6 using the binomial theorem is a fundamental skill in mathematics with wide-ranging applications.