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Question

The polynomial (x+y)^9 is expanded in decreasing powers of x. The second and third terms have equal values when evaluated at x=p and y=q, where p and q are positive numbers whose sum is one. What is the value of p?

Accepted Answer

The second and third term of (x+y)^9 is 9x^8y and 36x^7y^2, respectively. For them to be equal when x = p, p4 = y. For them to be equal when y = q, x = 4q. Then p+p4 = q+4q, so 5p4 = 5q, which simplifies to p = 4q. Since p+q = 1, p = 4/5.