Find the coefficient of the indicated term in the expansion of the given power of a binomial. Identify the missing exponents.
(x^7y^2);(x+y)^?
Please show work I need to learn how to do it.
you know that it must be (x+y)^9 to get a ?x^7y^2
so the term is
C(9,7) x^7 y^2
= 36x^7y^2
Observe that the 10th row of Pascal's triangle is
1 9 36 84 126 126 84 36 9 1
To find the coefficient of the indicated term in the expansion of the given binomial, you can use the Binomial Theorem. The Binomial Theorem provides a formula to expand a binomial raised to a power.
The Binomial Theorem states that for any positive integers n and k, the coefficient of the kth term in the expansion of (a + b)^n is given by:
C(n, k) * a^(n-k) * b^k
Where C(n, k) represents the binomial coefficient. The binomial coefficient is the number of ways to choose k items from a set of n items and is calculated using the formula:
C(n, k) = n! / (k! * (n-k)!)
In your case, we need to find the coefficient of the term x^7y^2 in the expansion of (x+y)^?. To determine the missing exponent, we can set up the following equation:
7 = (n-k)
2 = k
Solving this system of equations, we find that n = 9, k = 2. Therefore, we need to expand (x+y)^9 to find the desired coefficient.
Expanding (x+y)^9 using the Binomial Theorem:
(x+y)^9 = C(9, 0) * x^9 * y^0 + C(9, 1) * x^8 * y^1 + C(9, 2) * x^7 * y^2 + ...
Since we are interested in the coefficient of the term x^7y^2, we can ignore all other terms and focus on C(9, 2) * x^7 * y^2.
Calculating C(9, 2):
C(9, 2) = 9! / (2! * (9-2)!)
= 9! / (2! * 7!)
= (9 * 8) / (2 * 1)
= 36
Therefore, the coefficient of x^7y^2 in the expansion of (x+y)^9 is 36.
In summary, to find the coefficient of the indicated term in the expansion of a binomial:
1. Set up a system of equations to determine the missing exponents.
2. Use the Binomial Theorem formula: C(n, k) * a^(n-k) * b^k.
3. Expand the binomial using the Binomial Theorem.
4. Identify the coefficient of the desired term by calculating the binomial coefficient.