I need help trying to write out these binomial expansions: they all need to be raised to the 8th power. Thanks
1. (x + y) 2. (w + z) 3. (x - y) 4. (2a + 3b)
- Now explain how your answer for #1 could be used as a formula to help you answer each of the other items. In each case, for #2, 3 and 4, tell what would x equal and what would y equal.
- Without writing out the whole polynomial, find the 16th term is (x + y)19. Explain how you constructed it.
Recall your Pascal's triangle for the expansion of (a+b)^n
On the nth row, you have the kth element: C(n,k)a^(n-k+1)b^(k-1)
That is, on the 8th row you have
C(8,0)a^8b^0 C(8,1)a^7b^1 . . .
= a^8 + 8a^7b + 28a^6b^2 + ... + 28a^2b^6 + 8ab^7 + b^8
There are lots of places online where you can find Pascal's triangle.
So, for this problem, the 16th term of the 19th row is
C(19,15)(2a)^4(3b)^15 = 3876(2a)^4(3b)^15
= 3876(2^5)(3^14)a^4b^15
= 3876(16)(14348907)a^4b^15
= 889861816512a^4b^15
=
Oops. It was just (x+y)^19. My bad.
C(19,15)x^4(y^15 = 3876x^4y^15
To expand a binomial raised to the 8th power, we can use the binomial theorem. The binomial theorem states that the expansion of (a + b)^n can be obtained by summing the terms of the form (n choose k) * a^(n-k) * b^k, where (n choose k) is the binomial coefficient.
Let's apply this to each of the given binomials raised to the 8th power.
1. (x + y)^8:
Using the binomial theorem, we can find the expansion of (x + y)^8 as follows:
- For each term, the exponent of x decreases by 1, while the exponent of y increases by 1.
- The binomial coefficients can be found using Pascal's triangle or by using the formula (n choose k) = n! / (k! * (n-k)!).
2. (w + z)^8:
The same process can be applied here. Since we already have the expansion of (x + y)^8, we can use it as a formula to determine the expansion of (w + z)^8. In this case, we replace x with w and y with z.
3. (x - y)^8:
Similarly, we can use the expansion formula of (x + y)^8 and replace every occurrence of y with -y to get the expansion of (x - y)^8.
4. (2a + 3b)^8:
Once again, we can use the expansion formula of (x + y)^8. Here, we replace x with 2a and y with 3b.
Now let's answer the second part of the question:
To use the answer for #1 as a formula for #2, 3, and 4, we can substitute specific values for x and y in the derived formulas. For example:
- For #2, if x = w and y = z, we substitute these values into the derived formula to calculate the expansion of (w + z)^8.
- For #3, if x = x and y = -y, we substitute these values into the derived formula to calculate the expansion of (x - y)^8.
- For #4, if x = 2a and y = 3b, we substitute these values into the derived formula to calculate the expansion of (2a + 3b)^8.
Now let's move on to the last part of the question:
To find the 16th term of (x + y)^19 without writing out the whole polynomial, we can use the formula for the term of a binomial expansion:
(n choose k) * a^(n-k) * b^k, where n is the exponent, k is the term's position (starting from 0), a is the coefficient of the first term, and b is the coefficient of the second term.
In this case, n = 19, a = x, b = y, and k = 16. Substituting these values into the formula, we can construct the term: (19 choose 16) * x^(19-16) * y^16.