AMC 10 · 2004 · #17

Grade 11 algebra
Archetype Arithmetic Expression Decomposition └ Sub-archetype Symmetric Function of Roots via Vieta
vieta-formulaslogarithm-propertiespolynomial-roots convert-to-algebralogical-deduction ↑ Prerequisites: vieta-formulas
📏 Medium solution 💡 3 insights
Problem
A cubic with real coefficients has three different positive roots. The base-2 logarithms of those roots add up to 5. Determine the coefficient a.

Pick an answer.

(A)
-256
(B)
-64
(C)
-8
(D)
64
(E)
256
How to solve
Strategy Introduce a Variable

The roots are never going to be found individually - there is not enough information for that, and b is left free on purpose. So name them x₁,x₂,x₃ and hunt for the one combination the data actually controls. A sum of logarithms is a logarithm of a product, so the given condition is really a statement about x₁x₂x₃. The other place a product of roots shows up is the constant term of the factored form of the polynomial, so writing the same cubic in two ways - expanded and factored - puts the known quantity and the unknown a on opposite sides of one equation.

1STEP 1

Turn the log sum into a product

A sum of logs is the log of a product, so the roots multiply to 32.

log₂(x₁x₂x₃)=5 ⟹ x₁x₂x₃=2⁵=32
2STEP 2

Write the cubic in factored form

Distinct roots let the cubic be written fully factored.

8x³+4ax²+2bx+a=8(x-x₁)(x-x₂)(x-x₃)
3STEP 3

Read off the constant terms

Substituting zero reads the constant term straight off as -256.

a=-8x₁x₂x₃=-8 · 32=-256
4STEP 4

Confirm such a cubic exists

An explicit root triple shows such a cubic exists, so the answer is -256, choice (A).

x₁+x₂+x₃=-a/2=128, x₁x₂x₃=32: (32, 48+√(2303), 48-√(2303))
Answer
-256
Two quick filters settle the choice without redoing the work. Sign: the roots are positive, so their product is positive, and the factored form makes that product equal -a/8; hence a is negative, which eliminates 64 and 256 at once. Size: the product is exactly 2⁵=32, so |a|=8 · 32=256, which eliminates -64 and -8. Only -256 survives both. Step 4 adds the part that pure elimination cannot supply - an explicit cubic with three distinct positive roots - so the answer is not resting on an empty hypothesis.
💡Key takeaway

Adding logarithms is multiplying the numbers, and the constant term of a polynomial is the product of its roots up to sign - so the two facts meet on the product and hand you the answer.

  • Turn the log sum into a product
  • Write the cubic in factored form
  • Read off the constant terms
  • Confirm such a cubic exists

More like this

Same archetype — closest grade level first.