AMC 10 · 2017 · #24
Grade 8 algebraPick an answer.
AMC 10 2017 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
f shares all three roots of g and has exactly one more root. Since both are monic, f must equal g times one linear factor (x - r). Name that fourth root r, expand the product, then match it term-by-term against the given f. Matching coefficients turns the problem into small linear equations that pin down r, a, b, and c.
Write f as g times a linear factor
Both monic with f of degree 4, so f is g times one leftover factor: name the fourth root r and write f(x) = g(x)(x - r).
A monic quartic that contains a monic cubic's roots is just that cubic times one leftover (x - r).
6.EE.B.6Introduce A VariableExpand the product
Multiply out (x³ + ax² + x + 10)(x - r) and collect like powers, giving a quartic with coefficients in terms of a and r.
Distributing turns the factored form into one whose coefficients we can compare directly.
6.EE.A.3Introduce A VariableMatch coefficients to find r
Match the expansion to f: a - r = 1, 1 - ar = b, 10 - r = 100, -10r = c. Only the x term isolates r: 10 - r = 100 → r = -90.
Two equal polynomials must agree coefficient by coefficient, so each power gives its own little equation.
Two equal polynomials must agree coefficient by coefficient, so each power gives its own equation.
▸ Why?
Two expressions that agree for every input must match term by term.
▸ Why?
The factored form is valid because a polynomial vanishes exactly where one of its factors does.
Find a, c, then b
With r = -90: a = 1 + r = -89, c = -10r = 900, and b = 1 - ar = 1 - (-89)(-90) = -8009.
Once r is known, every other coefficient drops out by plugging back into its own equation.
7.NS.A.2Identify SubproblemsEvaluate f(1)
Sum f's coefficients: f(1) = 2 + b + 100 + c = 2 - 8009 + 100 + 900 = -7007, matching g(1)(1 - r) = (-77)(91) = -7007.
f(1) is just the sum of f's coefficients, and the factored form gives a clean second route to the same number.
6.EE.A.2Introduce A VariableWhen one monic polynomial holds all the roots of a smaller monic one, it is just the smaller one times a single (x - r) leftover, and matching coefficients reveals everything.
- Write f as g times a linear factor
- Expand the product
- Match coefficients to find r
- Find a, c, then b
- Evaluate f(1)