Find the coefficient of the fifth term in the expansion(rootK+3)^8

This is the way I remember it. Of course, you can use the formula (given there also), but jotting down the triangle is easy, and counting to row 8 is very easy. http://ptri1.tripod.com/#made

for (a+b)^n

termr+1 = C(n,r) a^(n-r) b^r

so for (√k + 3)^8
term(5) = C(8,4) (√k)^4 3^4
= 70 k^2 (81)
= 5670k^2

so the coefficient is 5670

To find the coefficient of a specific term in a binomial expansion, we can use the binomial theorem. The binomial theorem states that for any positive integers n and m, and any real number x and y, the expansion of the binomial (x + y)^n can be expressed as:

(x + y)^n = C(n, 0) * x^n * y^0 + C(n, 1) * x^(n-1) * y^1 + C(n, 2) * x^(n-2) * y^2 + ... + C(n, n) * x^0 * y^n

where C(n, k) represents the binomial coefficient, calculated using the formula:

C(n, k) = n! / (k! * (n-k)!)

In this case, we are given the expression (rootK + 3)^8. To find the coefficient of the fifth term, we need to determine the values of x, y, n, and k.

Let's analyze the given expression:

(rootK + 3)^8

Here, x = rootK and y = 3. Also, the exponent is 8.

We are looking for the coefficient of the fifth term, which corresponds to k = 5.

Now, we can use the binomial theorem formula to find the coefficient.

We know that C(n, k) represents the coefficient of the kth term in the expansion. So, in our case, we need to find C(8, 5).

Using the binomial coefficient formula:

C(n, k) = n! / (k! * (n-k)!)

Plugging in the values, we have:

C(8, 5) = 8! / (5! * (8-5)!)

Simplifying further:

C(8, 5) = 8! / (5! * 3!)

Now, we can calculate the coefficient by evaluating the factorial terms:

8! = 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1
5! = 5 * 4 * 3 * 2 * 1
3! = 3 * 2 * 1

Substituting the factorial terms into the formula:

C(8, 5) = (8 * 7 * 6 * 5 * 4 * 3 * 2 * 1) / ((5 * 4 * 3 * 2 * 1) * (3 * 2 * 1))

Evaluate the above expression:

C(8, 5) = 8 * 7 * 6 / (3 * 2 * 1)

C(8, 5) = 56

Therefore, the coefficient of the fifth term in the expansion (rootK + 3)^8 is 56.