AMC 10 · 2003 · #21

Grade 11 algebra
Archetype Arithmetic Expression Decomposition └ Sub-archetype Symmetric Function of Roots via Vieta
polynomial-rootsvieta-formulaspolynomial-factoring work-backwardsguess-and-check ↑ Prerequisites: polynomial-roots
📏 Long solution 💡 3 insights
Problem
The graph of P(x) = x⁵ + ax⁴ + bx³ + cx² + dx + e crosses the x-axis at five different points, and one of those points is the origin. Among the coefficients a, b, c, d, e, decide which one is impossible to be zero.

Pick an answer.

(A)
a
(B)
b
(C)
c
(D)
d
(E)
e
How to solve
Strategy Introduce a Variable

The conditions are all about roots while the question is all about coefficients, so the plan is to work backwards from the roots to the coefficients (Tool #11). Name the four unknown nonzero roots (Tool #4); then the root at 0 lets P be split as x times a quartic (Tool #7), and the quartic's constant term is exactly the coefficient d. That single subproblem settles d. But proving one coefficient is never zero is only half the job: the question asks which one cannot be zero, so the other candidates must be shown to be genuinely reachable. That means building explicit polynomials with a=0, with c=0, and with b=0 (Tool #6), and noticing that e is not merely allowed to be zero but forced to be. Together those rule out every other choice (Tool #3).

1STEP 1

The origin kills the constant term

Substituting the origin leaves only the constant, so e is always zero — not what is asked.

P(0)=0⁵+a · 0⁴+b · 0³+c · 0²+d · 0+e=e=0
2STEP 2

Split off the root at zero

Factoring out that root shifts the list, making d the new constant term.

P(x)=x(x⁴+ax³+bx²+cx+d)=x Q(x)
3STEP 3

d is a product of four nonzero numbers

That constant equals the product of four nonzero roots, so d is never zero.

d=Q(0)=r₁r₂r₃r₄ ≠ 0
4STEP 4

Show the other three really can vanish

Explicit examples make a, b and c each vanish, so none of them qualifies.

x(x²-1)(x²-4)=x⁵-5x³+4x; x(x-1)(x+1)(x-2)(x-1/2)=x⁵-5/2x⁴+5/2x²-x
5STEP 5

Put the five together

Exactly one coefficient survives, so the answer is d, choice (D).

e=0 always; a=b=c=0 each achievable; d=r₁r₂r₃r₄ ≠ 0
Answer
d
Read the conclusion off the graph instead of the algebra. Near x=0 the polynomial P(x)=x⁵+…+dx behaves like the line y=dx, because the higher powers shrink much faster. If d were 0 the curve would flatten against the x-axis at the origin and touch rather than cross, which means x² would divide P and the origin would be a repeated root — impossible when all five intercepts are distinct. So d ≠ 0 is exactly the statement that the curve genuinely cuts through the origin. Both witness polynomials confirm it numerically: x⁵-5x³+4x has d=4 and roots 0,± 1,± 2, and x⁵-5/2x⁴+5/2x²-x has d=-1 and roots 0,1,-1,2,1/2. Both are five distinct real numbers including 0, as required, so the examples are legitimate and not degenerate.
💡Key takeaway

A curve that cuts straight through the origin must have a nonzero slope there, and that slope is the coefficient of x — which is why d is the one coefficient that can never disappear.

  • The origin kills the constant term
  • Split off the root at zero
  • d is a product of four nonzero numbers
  • Show the other three really can vanish
  • Put the five together

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