1979 AHSME Problems/Problem 25
Contents
[hide]Problem 25
If and are the quotient and remainder, respectively, when the polynomial is divided by , and if and are the quotient and remainder, respectively, when is divided by , then equals
Solution
Solution by e_power_pi_times_i
First, we divide by using synthetic division or some other method. The quotient is , and the remainder is . Then we plug the solution to into the quotient to find the remainder. Notice that every term in the quotient, when , evaluates to . Thus .
Solution 2
Using the remainder theorem, we see that the remainder upon dividing by is equal to Thus, we have that Isolating we see Using the difference of squares factorization repeatedly, we get Finally, plugging in again, the final answer is
See also
1979 AHSME (Problems • Answer Key • Resources) | ||
Preceded by Problem 24 |
Followed by Problem 26 | |
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