Polynomials path · A04
Before this lesson: Remainder and factor theorems, Complex numbers: optional bridge
Your goal: Use the theorem with the correct coefficient field and multiplicities.
Start with the idea and one example. Take a break before the written task if you need it. The questions are original teaching exercises, not official past-paper questions.
Polynomial roots and multiplicity: the key idea
The fundamental theorem of algebra states that every nonconstant complex-coefficient polynomial has a complex root. Repeated factorisation therefore gives exactly n complex roots counted with multiplicity for a degree-n polynomial. This theorem is accepted here; a full proof lies beyond this elementary course. A separate elementary consequence of the factor theorem is that a nonzero degree-n polynomial has at most n distinct roots. Thus two polynomials of degree at most n that agree at n+1 distinct points are identical. Count multiplicities and distinct roots separately.
A worked example
How many distinct roots and how many roots counted with multiplicity does (x−2)³(x+1)² have?
The distinct roots are 2 and −1, so there are two. Their multiplicities are 3 and 2 respectively, summing to 5, the degree. Repeated copies count for the theorem even though they mark only two points.
Your turn: change one thing
Polynomials P and Q have degree at most 4 and agree at five distinct real inputs. Prove P=Q.
Try this on paper before opening the explanation.
Compare your reasoning
P−Q has degree at most 4. If nonzero, it could have at most four distinct roots. The five agreement points are five roots, so P−Q must be zero.
Pause and check
A trap to avoid: Assuming every polynomial has a real root; present the theorem as accepted, not a purported elementary proof.
Practise and adjust the level
Foundation checks the language; Core applies the method; Stretch asks you to choose or justify an idea. These are levels within this lesson. A session selects six of the nine questions; unused questions allow the level to change. Advanced theory still needs written practice.
Interactive practice loads here. You can also use the complete question set below.
Write a complete argument
Prove the distinct-root bound using the factor theorem.
Planning hint
List the assumptions and the exact conclusion. Identify the definition or theorem in this lesson that connects them. Explain why its conditions hold before using it.
Read the full solution after your attempt
Induct on degree. A nonzero constant has no root. If a degree-n polynomial has root a, factor it as (x−a)Q with Q of degree n−1. Every other distinct root b satisfies 0=(b−a)Q(b) and b−a≠0, so Q(b)=0. By induction there are at most n−1 such roots; including a gives at most n.
My proof notebook
Write on paper, or keep a draft here. Compare your reasoning with the solution only after a real attempt. The checklist is your own review, not an automatic mark.
All nine practice questions
Prefer paper or have JavaScript switched off? The complete question set, hints and solutions are here. Interactive practice uses these same questions in an order chosen from your answers.
1. A degree-5 complex polynomial has how many complex roots counted with multiplicity?
Foundation
- 6
- 5
- 0
- 4
Hint
Use the theorem’s counting convention.
Answer and reasoning
5. The number counted with multiplicity equals the degree.
2. How many distinct roots does (x−1)⁴ have?
Foundation
- 2
- 4
- 1
- 0
Hint
All four factors vanish at the same input.
Answer and reasoning
1. The only root is 1; it has multiplicity four.
3. What is the maximum number of distinct roots of a nonzero cubic?
Core
- 3
- 2
- 4
- 6
Hint
Use the distinct-root bound.
Answer and reasoning
3. A degree-3 nonzero polynomial has at most three distinct roots.
4. Does the fundamental theorem guarantee a real root of every real polynomial?
Core
- Yes, for every quadratic
- No; x²+1 is a counterexample
- Yes, always
- Only if the polynomial is monic
Hint
The theorem guarantees complex roots.
Answer and reasoning
No; x²+1 is a counterexample. x²+1 has roots ±i but no real root.
5. P has degree at most 3 and P(0)=P(1)=P(2)=P(3)=7. What follows?
Stretch
- P has four distinct complex roots
- P(4)=0
- P is the constant polynomial 7
- P has degree exactly 3
Hint
Apply the root bound to P−7.
Answer and reasoning
P is the constant polynomial 7. P−7 has four distinct roots and degree at most 3, so it is the zero polynomial.
6. What extra condition is essential when using n+1 values to identify a degree-at-most-n polynomial?
Stretch
- The values are all positive
- The inputs are all prime
- The polynomial is monic
- The inputs are distinct
Hint
Repeated evaluation at one input adds no new root.
Answer and reasoning
The inputs are distinct. The root-count argument needs n+1 distinct roots of the difference.
7. Multiplicity counts what?
Foundation
- How many times a root’s linear factor occurs
- Only distinct roots
- Only positive roots
- The number of coefficients
Hint
Repeated factors contribute repeated roots.
Answer and reasoning
How many times a root’s linear factor occurs. A factor gives root a multiplicity m.
8. For (x−1)³(x+2), how many distinct roots occur?
Core
- 1
- 2
- 4
- 3
Hint
List the root values.
Answer and reasoning
2. The distinct roots are 1 and −2, though total multiplicity is four.
9. If a polynomial of degree at most 4 vanishes at five distinct points, what follows?
Stretch
- It has degree 5
- It is always x⁴
- It has no coefficients
- It is the zero polynomial
Hint
Use the bound on distinct roots.
Answer and reasoning
It is the zero polynomial. A nonzero degree-at-most-four polynomial cannot have five distinct zeros.
Choose your next step
Continue to Solving polynomial equations. If this felt difficult, return to a prerequisite above. Every lesson stays open.
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