Consider an expression of the form (ax+by)^N and find the values of a, b, and N. Recall that the sum of the exponents in each term is equal to N. You can assign a particular number to a and use Pascal's Triangle to find b.
Example Solution: (x+ 65y)^5
Practice makes perfect
We are interested in writing a power of a binomial for which the second term of the expansion is 6x^4y.
(A+B)^N = ? + 6x^4y + ? + ⋯ + ?
Our job is to determine A, B, and N. Since the sum of the exponents in each term is N and the second term has exponents 4 and 1, we conclude that N=4+1=5.
(A+B)^5 = ? + 6x^4y + ? + ⋯ + ?The variables in the second term are x and y, and these are the variables that must be inside the parentheses on the left-hand side. However, they can be multiplied by a constant. Thus, we will call A=ax and B=bx.
(ax + bx)^5 = ? + 6x^4y + ? + ⋯ + ?
To determine the values of a and b, we will check the fifth row in Pascal's Triangle.
As we can see, the second coefficient in the fifth row is 5. Thus, in the expansion of (ax+by)^5 the second term must have the following form.
Second Term: 5(ax)^4(by)
Let's equate the expression above and the expression for the second term that we have been given.
5(ax)^4(by) = 6x^4y
For simplicity, let's suppose that a=1 and substitute it above to find the value of b.
In consequence, the power of the binomial we were looking for is the one shown below.
(x+6/5y)^5 = ? + 6x^4y + ? + ⋯ + ?
Keep in mind that the expression above is just an example and your answer may vary.
Extra
Note
Although your answer may be different the exponent has to be 5, the variables inside the parentheses must be x and y, and the coefficients of x and y must be positive. The different parts in your expression can be only the coefficients next to x and y.
