Question

If we wish to expand (x + y) 8 , what is the coefficient of x 5 y 3 ? what is the coefficient of x 3 y 5 ?

Answer 1

To expand (x + y)^8, we can use the binomial theorem. The coefficient of x^5y^3 is 56, and the coefficient of x^3y^5 is also 56.

Step-by-step explanation:

To expand (x + y)8, we can use the binomial theorem. The binomial theorem states that for any positive integer n, the expansion of (a + b)n can be found using the formula:

(a + b)n = C(n, 0) anb0 + C(n, 1) an-1b1 + C(n, 2) an-2b2 + ... + C(n, n) a0bn.

In this case, since we are expanding (x + y)8, the coefficient of x5y3 can be found by plugging in n = 8, and k = 5 and j = 3 into the formula C(n, k) an-kbk, which gives us C(8, 5) x8-5y5 = 56 x3y5. Similarly, the coefficient of x3y5 can be found using the same formula, plugging in k = 3 and j = 5, which gives us C(8, 3) x8-3y3 = 56 x5y3.

Answer 2

We use the binomial theorem to answer this question. Suppose we have a trinomial (a + b)ⁿ, we can determine any term to be:

[n!/(n-r)!r!] a^(r) b^(n-r)

a.) For x⁵y³, the variables are: x=a and y=b. We already know the exponents of the variables. So, we equate this with the form of the binomial theorem.

r = 5
n - r = 3
Solving for n,
n = 3 + 5 = 8

Therefore, the coefficient is equal to:
Coefficient = n!/(n-r)!r! = 8!/(8-5)!8! = 56

b.) For x³y⁵, the variables are: x=a and y=b. We already know the exponents of the variables. So, we equate this with the form of the binomial theorem.

r = 3
n - r = 5
Solving for n,
n = 5 + 3 = 8

Therefore, the coefficient is equal to:
Coefficient = n!/(n-r)!r! = 8!/(8-3)!8! = 56