I am a little confused with the term expand. the question is find each term: the fourth term in the expansion of (4y+X)^4
I looked at pascals triangle is the answer -4X(-4y^3)
does the problem call for the renaming or rearranging of x and y?
ah ham you know Pascal's triangle so you know binomial theorem after all
now
call a = 4 y
call b = x
then
(a+b)^4 = a^4 + 4 a^3b + 6 a^2b^2 + 4 a b^3 + b^4
which is
256 y^4 + 256x +6 (4y)^2 x^2 etc
To find the fourth term in the expansion of (4y + x)^4, you need to use the binomial theorem. The binomial theorem allows us to expand a binomial raised to a certain power.
The formula for the binomial expansion is:
(a + b)^n = C(n,0) * a^n * b^0 + C(n,1) * a^(n-1) * b^1 + C(n,2) * a^(n-2) * b^2 + ... + C(n,n-1) * a^1 * b^(n-1) + C(n,n) * a^0 * b^n
In this case, your binomial is (4y + x), and you want to find the fourth term.
Using the formula, we can see that the fourth term will be:
C(4,3) * (4y)^1 * (x)^(4-1) = 4 * (4y) * x^3 = 16xy^1 * x^3 = 16xy^1 * x^3 = 16xy^1x^3 = 16x^4y
So, the fourth term in the expansion of (4y + x)^4 is 16x^4y.
Regarding your question about renaming or rearranging x and y, it doesn't require renaming or rearranging. The binomial theorem allows you to directly expand the expression as it is given.