Stu Dent has mastered a new trick. It goes as follows...
First: Stu asks someone to choose a random poylnomial p(x), of any degree, with nonnegative integer coefficients.
Second: Stu states that he has an amazing mathemagical power in that he can determine the person's polynomial after knowing just two seemingly unconnected values [i.e. he will provide a value n and the person tells Stu the value p(n)].
Task: Try to figure out what two values of n Stu provides to learn p(n)?
Note: Stu thanks I.B. Keene (a magical name in itself) from the University of Michigan for being the creator of this trick.
Source: "A Perplexing Polynomial Puzzle," College Mathematics Journal, March 2005, p. 100
Hint: For example, suppose you have a hidden polynomial.
Stu gives you n=1, and you reply that p(1) = 14.
Stu now gives you n=15, and you reply p(15) = 4074.
Stu replies that the polynomial must be p(x)=x3+3x2+x+9.
And, Stu is correct...and we all clap in wonder!!
Solution Commentary: It would be too easy to just reveal the magic....rather, I am going to give you two more examples....from them, try to determine the method behind Stu's magic...
Example 2:
Stu gives you n=1, and you reply that p(1) = 12.
Stu now gives you n=13, and you reply p(13) = 818,701,080.
Stu replies that the polynomial must be p(x)=x8+8x5+x+2.
And, Stu is correct...and we all clap in wonder!!
Example 3:
Stu gives you n=1, and you reply that p(1) = 9.
Stu now gives you n=10, and you reply p(10) = 153.
Stu replies that the polynomial must be p(x)=x2+5x+3.
And, Stu is correct...and we all clap in wonder!!
Study this last example...it almost gives away Stu's magic!
