%5E4-expand.jpeg "(2x+1)^4 Expand")
Expanding (2x + 1)^4
Expanding expressions of the form (ax + b)^n, where n is a positive integer, can be a tedious process. However, there are a couple of methods that make it easier.
1. Binomial Theorem
The binomial theorem provides a formula for expanding any binomial raised to a power:
(a + b)^n = ∑ (n choose k) * a^(n-k) * b^k
where:
- n is the power
- k ranges from 0 to n
- (n choose k) is the binomial coefficient, calculated as n! / (k! * (n-k)!)
Let's apply this to (2x + 1)^4:
- a = 2x
- b = 1
- n = 4
Expanding using the formula:
(2x + 1)^4 = (4 choose 0) * (2x)^4 * 1^0 + (4 choose 1) * (2x)^3 * 1^1 + (4 choose 2) * (2x)^2 * 1^2 + (4 choose 3) * (2x)^1 * 1^3 + (4 choose 4) * (2x)^0 * 1^4
Calculating the binomial coefficients:
- (4 choose 0) = 1
- (4 choose 1) = 4
- (4 choose 2) = 6
- (4 choose 3) = 4
- (4 choose 4) = 1
Substituting the values and simplifying:
(2x + 1)^4 = 1 * 16x^4 * 1 + 4 * 8x^3 * 1 + 6 * 4x^2 * 1 + 4 * 2x * 1 + 1 * 1 * 1
Finally, we get:
(2x + 1)^4 = 16x^4 + 32x^3 + 24x^2 + 8x + 1
2. Pascal's Triangle
Pascal's Triangle provides a visual representation of the binomial coefficients. Each row represents the coefficients for a different power of a binomial.
To expand (2x + 1)^4, we need the 5th row of Pascal's Triangle:
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
The coefficients are 1, 4, 6, 4, and 1. Now, we use these coefficients, following the pattern of the binomial theorem:
(2x + 1)^4 = 1(2x)^4 + 4(2x)^31 + 6(2x)^21^2 + 4(2x)1^3 + 11^4**
Simplifying:
(2x + 1)^4 = 16x^4 + 32x^3 + 24x^2 + 8x + 1
Both methods lead to the same result:
(2x + 1)^4 = 16x^4 + 32x^3 + 24x^2 + 8x + 1