AMC 10 · 2017 · #24

Grade 8 algebra
Archetype Coordinate Plane Figure Computation └ Sub-archetype Figure Constraint Equations for Unknown Quantity
polynomial-rootspolynomial-factoringvieta-formulas polynomial-factoringvieta-formulas ↑ Prerequisites: polynomial-roots
📏 Medium solution 💡 3 insights
Problem
A cubic g(x) = x³ + ax² + x + 10 has three distinct roots, and every one of those roots is also a root of the quartic f(x) = x⁴ + x³ + bx² + 100x + c. Find the value of f(1).

Pick an answer.

(A)
-9009
(B)
-8008
(C)
-7007
(D)
-6006
(E)
-5005

AMC 10 2017 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Introduce a Variable

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.

1STEP 1

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).

f(x) = (x³ + ax² + x + 10)(x - r)
2STEP 2

Expand 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.

f(x) = x⁴ + (a - r)x³ + (1 - ar)x² + (10 - r)x - 10r
3STEP 3

Match 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.

10 - r = 100 → r = -90
4STEP 4

Find a, c, then b

With r = -90: a = 1 + r = -89, c = -10r = 900, and b = 1 - ar = 1 - (-89)(-90) = -8009.

a = -89, c = 900, b = 1 - (-89)(-90) = -8009
5STEP 5

Evaluate 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) = 2 - 8009 + 100 + 900 = -7007
Answer
-7007
Two independent routes agree: summing coefficients gives 2 - 8009 + 100 + 900 = -7007, and the factored form gives g(1)(1 - r) = (-77)(91) = -7007. The factored route never needs b at all, so it cross-checks the coefficient matching. -7007 is exactly choice (C).
💡Key takeaway

When 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)

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