Problem 25 · 2023 Math Kangaroo
Stretch
Algebra & Patterns
factorizationsum-constraint
A part of a polynomial of degree five is illegible due to an ink stain (see diagram). It is known that all zeros of the polynomial are integers. What is the highest power of \(x - 1\) that divides this polynomial?
Show answer
Answer: D — \((x-1)^4\)
Show hints
Hint 1 of 2
Vieta's formulas link the visible coefficients to the sum and product of the roots.
Still stuck? Show hint 2 →
Hint 2 of 2
All roots are integers, the product is 7 and the sum is 11 — that pins them down.
Show solution
Approach: recover the integer roots with Vieta's formulas
- For x5 − 11x4 + ... − 7, the integer roots have product 7 and sum 11.
- The only integer multiset is 7, 1, 1, 1, 1 (product 7, sum 11).
- So (x−1) appears four times, and the highest power dividing it is (x−1)4.
Mark:
· log in to save