AMC 10 · 2018 · #22
Grade 11 algebraPick an answer.
Evaluating at x = -1 turns the condition into one linear equation in the four digits, but two of them arrive with minus signs and that blocks every standard counting formula. The fix is to look at those two digits from the other end: replace a by how far it falls short of 9. Since the digits 0 through 9 are symmetric about that flip, the swap loses nothing, and it converts the mixed-sign equation into a plain "nonnegative numbers adding to a fixed total" problem, which stars and bars counts directly.
Name the four coefficients
Name the four coefficients.
A polynomial is nothing more than its list of coefficients, so counting polynomials means counting lists.
9.A-SSE.A.1Introduce A VariableEvaluate the polynomial at -1
Substituting makes the signs alternate.
Plugging in -1 makes every power equal ± 1, so the polynomial becomes a plain alternating sum of its coefficients.
Plugging in minus one makes every power equal plus or minus one, so the polynomial becomes an alternating sum.
▸ Why?
An even power of minus one is one and an odd power is minus one, so the signs alternate down the list.
▸ Why?
Multiplying a coefficient by one leaves it alone, so what remains is the coefficients themselves.
See what is blocking the count
Mixed signs block a direct count.
Mixed plus and minus signs are exactly what stops a sum equation from being a simple sharing-out problem.
9.A-CED.A.3Organize Information In More WaysFlip the two negative digits
Flip the negative digits by subtracting from nine.
Counting how far a falls short of 9 instead of how far it rises above 0 turns its minus sign into a plus sign.
9.A-SSE.A.2Change Focus Count The ComplementCheck the swap loses nothing
The swap loses nothing.
A relabelling you can undo never changes how many things there are.
9.F-IF.A.1Organize Information In More WaysConfirm the digit ceiling never binds
The digit ceiling never binds.
You cannot overspend a budget of 9 when every share is nonnegative.
9.A-CED.A.3Extreme PrincipleShare 9 units among 4 slots
Sharing nine among four slots gives 220.
Splitting a fixed total among a fixed number of slots is decided entirely by where the dividers go.
11.S-CP.B.9Make A Systematic ListWhen a sum equation has minus signs in it, measure those variables from the top instead of the bottom — use 9 - a in place of a — and the whole thing becomes a simple problem of sharing out a fixed total.
- Name the four coefficients
- Evaluate the polynomial at -1
- See what is blocking the count
- Flip the two negative digits
- Check the swap loses nothing
- Confirm the digit ceiling never binds
- Share 9 units among 4 slots