Polynomial Functions: Zeros, Multiplicity, and End Behavior
Why some zeros "bounce" off the x-axis while others cross — and how the leading term decides what happens far from zero.
Reading a polynomial from its graph
Three things you can read from any polynomial's graph: number of zeros, multiplicity of each zero, and end behavior. Combined, they often tell you the polynomial up to a leading coefficient.
End behavior is decided by the leading term
For P(x) = aₙxⁿ + (lower terms): as |x| → ∞, only aₙxⁿ matters.
- Even degree, positive leading: both ends rise (∪)
- Even degree, negative leading: both ends fall (∩)
- Odd degree, positive leading: falls left, rises right (∕)
- Odd degree, negative leading: rises left, falls right (∖)
Multiplicity changes how the graph meets the x-axis
If P(x) has factor (x − r)ᵐ, the zero at r has multiplicity m.
- Odd multiplicity (1, 3, 5, ...): the graph CROSSES the x-axis at r.
- Even multiplicity (2, 4, ...): the graph TOUCHES (bounces off) the x-axis at r.
For P(x) = (x − 1)(x + 2)²(x − 5)³ the zeros are 1 (mult 1, crosses), −2 (mult 2, bounces), 5 (mult 3, crosses). Total multiplicity = 6 = degree of P.
Fundamental Theorem of Algebra
A polynomial of degree n has exactly n complex roots counted with multiplicity. Real roots are a subset; complex roots come in conjugate pairs when coefficients are real.
Check yourself
Practice with real CBE questions
Try polynomial questions in Pre-Calc Sem A practice.