%5E6-expand.jpeg "(x+3)^6 Expand")
Expanding (x + 3)^6 Using the Binomial Theorem
The binomial theorem provides a systematic way to expand expressions of the form (x + y)^n. Here's how to apply it to expand (x + 3)^6:
Understanding the Binomial Theorem
The binomial theorem states:
(x + y)^n = ∑ (n choose k) * x^(n-k) * y^k
where:
- n is the power of the binomial
- k ranges from 0 to n
- (n choose k) represents the binomial coefficient, calculated as n! / (k! * (n-k)!)
Applying the Theorem to (x + 3)^6
- Identify n: In our case, n = 6.
- Calculate binomial coefficients: We need to find (6 choose k) for k = 0, 1, 2, 3, 4, 5, and 6:
- (6 choose 0) = 1
- (6 choose 1) = 6
- (6 choose 2) = 15
- (6 choose 3) = 20
- (6 choose 4) = 15
- (6 choose 5) = 6
- (6 choose 6) = 1
- Substitute into the binomial theorem formula:
(x + 3)^6 = (6 choose 0) * x^6 * 3^0 + (6 choose 1) * x^5 * 3^1 + (6 choose 2) * x^4 * 3^2 + (6 choose 3) * x^3 * 3^3 + (6 choose 4) * x^2 * 3^4 + (6 choose 5) * x^1 * 3^5 + (6 choose 6) * x^0 * 3^6
- Simplify:
(x + 3)^6 = x^6 + 18x^5 + 135x^4 + 540x^3 + 1215x^2 + 1458x + 729
Conclusion
By applying the binomial theorem, we have successfully expanded (x + 3)^6 to obtain the polynomial x^6 + 18x^5 + 135x^4 + 540x^3 + 1215x^2 + 1458x + 729.