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## Doing Magic With Polynomials

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!