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)=x^{3}+3x^{2}+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)=x^{8}+8x^{5}+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)=x^{2}+5x+3.

And, Stu is correct...and we all clap in wonder!!

Study this last example...it almost gives away Stu's magic!