AMC 10 · 2007 · #18
Grade 11 algebraPick an answer.
The finish line is handed to us and the start is hidden: we know where f vanishes and want its coefficients, which is Tool #11 (Work Backwards) exactly. Tool #4 (Introduce a Variable) names the two missing roots r and s so that 'real coefficients' can be turned into an equation about them instead of a slogan. The step the whole problem actually turns on is not finding the conjugates, which is routine; it is showing that four roots pin a monic quartic down uniquely. That needs the four numbers to be pairwise distinct, and it is worth noticing why: if the two given roots had been 2i and -2i, conjugation would have produced nothing new, f would not be determined, and the question would have no answer. Once f is pinned down, Tool #7 (Identify Subproblems) does the multiplication in the cheap order, pairing each root with its own conjugate so that two real quadratics appear and the product is visibly real. Tool #15 (Organize Information in More Ways) supplies the independent check: a + b + c + d = f(1) - 1, so the whole answer can be read off one evaluation of f at x = 1 using moduli, without ever expanding the polynomial.
Real coefficients force conjugate roots
Real coefficients bring in the two conjugate roots.
Reflecting the whole complex plane across the real axis leaves a real-coefficient polynomial looking exactly the same, so it must carry its zeros along in mirror pairs.
Because the coefficients are real, every nonreal root drags its mirror image along as another root.
▸ Why?
A real-coefficient polynomial has its nonreal roots in matched conjugate pairs.
▸ Why?
Reflecting the plane across the real axis leaves such a polynomial looking exactly the same.
Four distinct roots pin f down
Four roots and a matching degree pin the polynomial exactly.
Each new root that is different from the old ones forces out one more linear factor, and four linear factors already fill a quartic to the brim.
11.A-APR.B.2Work BackwardsPair each root with its conjugate
Pairing conjugates gives two real quadratics.
A number times its mirror image loses all trace of i, so grouping conjugates first is what turns complex bookkeeping into ordinary algebra.
11.N-CN.A.2Identify SubproblemsMultiply out and read the coefficients
Multiplying out and adding gives 9.
Once the polynomial is written out in standard form, the coefficients are just sitting there waiting to be added.
9.A-APR.A.1Identify SubproblemsConfirm by evaluating at x = 1
Evaluating at one confirms 9 without expanding, choice (D).
Plugging in x=1 turns every power into 1, so the sum of all the coefficients is just one function value in disguise.
9.A-SSE.A.2Organize Information In More WaysReal coefficients make roots come in mirror pairs, so two given roots become four; four different roots fill a quartic completely, and then x=1 hands you the sum of the coefficients in one shot.
- Real coefficients force conjugate roots
- Four distinct roots pin f down
- Pair each root with its conjugate
- Multiply out and read the coefficients
- Confirm by evaluating at x = 1