how do you expand the power of a binomal in (x+3)^4
To expand the power of a binomial, such as (x+3)^4, you can use the binomial theorem or the method of multiplying out. I will explain both methods:
Method 1: Binomial Theorem
The binomial theorem provides a formula for expanding a binomial raised to a power. According to the binomial theorem, the expansion of (x+3)^4 can be calculated using the following formula:
(x+3)^4 = C(4,0) * x^4 * 3^0 + C(4,1) * x^3 * 3^1 + C(4,2) * x^2 * 3^2 + C(4,3) * x^1 * 3^3 + C(4,4) * x^0 * 3^4
Here, C(n,r) represents the combination function, which calculates the number of ways to choose r items from a set of n items. The combination function can be calculated using the formula C(n,r) = n! / (r! * (n-r)!), where "!" denotes factorial.
Using this formula, you can expand (x+3)^4 as follows:
(x+3)^4 = C(4,0) * x^4 * 3^0 + C(4,1) * x^3 * 3^1 + C(4,2) * x^2 * 3^2 + C(4,3) * x^1 * 3^3 + C(4,4) * x^0 * 3^4
= 1 * x^4 * 1 + 4 * x^3 * 3 + 6 * x^2 * 9 + 4 * x^1 * 27 + 1 * x^0 * 81
= x^4 + 12x^3 + 54x^2 + 108x + 81
Method 2: Multiplying Out
Another way to expand (x+3)^4 is by multiplying out all the terms. Here's how you can do that:
(x+3)^4 = (x+3) * (x+3) * (x+3) * (x+3)
Start by multiplying the first two terms:
(x+3) * (x+3) = x * x + x * 3 + 3 * x + 3 * 3
= x^2 + 3x + 3x + 9
= x^2 + 6x + 9
Next, multiply the result by the third term:
(x^2 + 6x + 9) * (x+3) = x^2 * x + x^2 * 3 + 6x * x + 6x * 3 + 9 * x + 9 * 3
= x^3 + 3x^2 + 6x^2 + 18x + 9x + 27
= x^3 + 9x^2 + 27x + 27
Finally, multiply the result by the fourth term:
(x^3 + 9x^2 + 27x + 27) * (x+3) = x^3 * x + x^3 * 3 + 9x^2 * x + 9x^2 * 3 + 27x * x + 27x * 3 + 27 * x + 27 * 3
= x^4 + 3x^3 + 9x^3 + 27x^2 + 27x^2 + 81x + 27x + 81
= x^4 + 12x^3 + 54x^2 + 108x + 81
So, by multiplying out all the terms, we obtain the same result: (x+3)^4 = x^4 + 12x^3 + 54x^2 + 108x + 81.
Either method can be used to expand the power of a binomial, and both will give you the correct expansion result.