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Learn · Pre-Calculus

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.

7 minTEKS 3A-3DPre-Calculus

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 depends on degree (even/odd) & leading coefficient (±)Even, a > 0both ends ↑Even, a < 0both ends ↓Odd, a > 0↓ left, ↑ rightOdd, a < 0↑ left, ↓ rightquick mental model: for x → ±∞, leading term dominates

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 (∖)
Polynomial roots: P(x) = 0.1(x + 4)(x + 1)(x − 2) has zeros at x = −4, −1, 2−4−3−2−11232−2x = −4x = −1x = 2Factored form P(x) = a(x − r₁)(x − r₂)(x − r₃) reveals roots directly · Each linear factor = one zero

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.
🎯 Sum of multiplicities = degree

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.

Multiplicity: odd = crosses; even = bounces (tangent to axis)Multiplicity 1crosses straight throughMultiplicity 2bounces off (tangent)Multiplicity 3crosses with flat inflection

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

📌 Quick check
P(x) = (x + 1)²(x − 3). How does the graph meet the x-axis at x = −1?

Practice with CBE-style practice questions

Try polynomial questions in Pre-Calc Sem A practice.