%5E5.jpeg "(3x+4y)^5")
Expanding (3x+4y)^5 using the Binomial Theorem
The binomial theorem provides a powerful method for expanding expressions of the form (a + b)^n. Let's apply it to (3x + 4y)^5.
The Binomial Theorem
The binomial theorem states that for any real numbers a and b and any non-negative integer n:
(a + b)^n = (n choose 0)a^n b^0 + (n choose 1)a^(n-1)b^1 + (n choose 2)a^(n-2)b^2 + ... + (n choose n-1)a^1 b^(n-1) + (n choose n)a^0 b^n
Where (n choose k) represents the binomial coefficient, calculated as:
(n choose k) = n! / (k! * (n-k)!)
Applying the Theorem to (3x + 4y)^5
- Identify a and b: In our case, a = 3x and b = 4y.
- Identify n: n = 5.
Now, let's apply the binomial theorem:
(3x + 4y)^5 = (5 choose 0)(3x)^5(4y)^0 + (5 choose 1)(3x)^4(4y)^1 + (5 choose 2)(3x)^3(4y)^2 + (5 choose 3)(3x)^2(4y)^3 + (5 choose 4)(3x)^1(4y)^4 + (5 choose 5)(3x)^0(4y)^5
Calculating the Binomial Coefficients
- (5 choose 0) = 5! / (0! * 5!) = 1
- (5 choose 1) = 5! / (1! * 4!) = 5
- (5 choose 2) = 5! / (2! * 3!) = 10
- (5 choose 3) = 5! / (3! * 2!) = 10
- (5 choose 4) = 5! / (4! * 1!) = 5
- (5 choose 5) = 5! / (5! * 0!) = 1
Expanding the Expression
Substituting the calculated binomial coefficients and simplifying:
(3x + 4y)^5 = 1 * 243x^5 * 1 + 5 * 81x^4 * 4y + 10 * 27x^3 * 16y^2 + 10 * 9x^2 * 64y^3 + 5 * 3x * 256y^4 + 1 * 1 * 1024y^5
**(3x + 4y)^5 = ** 243x^5 + 1620x^4y + 4320x^3y^2 + 5760x^2y^3 + 3840xy^4 + 1024y^5
Conclusion
Therefore, the expanded form of (3x + 4y)^5 is 243x^5 + 1620x^4y + 4320x^3y^2 + 5760x^2y^3 + 3840xy^4 + 1024y^5. This process demonstrates the usefulness of the binomial theorem for expanding complex expressions.