Problem 9 · Factor an integer polynomial using known roots · 5 points
Made by AI and not reviewed by a real person.
A lock shows codes 3, 10, 23, 42, 67 for keys 1, 2, 3, 4, 5. Its hidden program is a polynomial with integer coefficients, but its degree is unknown; which code could key 6 show?
- 138
- 158
- 122
- 128
- 218
Show a hint
- The simplest formula fitting the five codes need not be the hidden program. What must every possible change to that formula have in common?
- Find a small polynomial that fits, then consider the roots of the difference between it and the hidden polynomial.
Show the answer and solution
Answer: E — 218
The five codes fit B(x)=3x²−2x+2, which gives B(6)=98. For any permitted program P, the integer-coefficient polynomial P−B has roots 1, 2, 3, 4, 5. Hence P(x)−B(x)=(x−1)(x−2)(x−3)(x−4)(x−5)Q(x) for an integer-coefficient polynomial Q. At key 6, the code must therefore be 98+120Q(6). Only 218 fits, and taking Q(x)=1 shows it is possible. The tempting 158 differs from 98 by 60, which passes separate divisibility checks against the five earlier keys, but those checks miss the required factor of 120.