Real Zeros, Multiplicity, and Graphs of Polynomial Functions

How multiplicities decide whether a graph crosses or touches the x-axis

Overview

This tutorial explains how the real zeros of a polynomial and their multiplicities determine where and how its graph meets the x-axis. The examples assume the polynomial is already in factored form; if yours isn't, see factoring polynomials.

For more advanced topics, see finding zeros of polynomials and the interactive polynomial grapher.

1. Zeros of a Polynomial

Let \(P(x)\) be a polynomial. The zeros are all values of \(x\) for which \(P(x) = 0\). Real zeros correspond to the \(x\)-intercepts of the graph — where the curve either crosses the x-axis or touches it and turns around.

Example 1 — Find the zeros

Find the zeros of \( P(x) = 5(x - 1)(x - 2) \).

Step-by-step solution

Set the polynomial equal to zero: \( 5(x - 1)(x - 2) = 0 \).

By the zero product property, each factor must be zero:

\[ x - 1 = 0 \quad \text{or} \quad x - 2 = 0 \]

Answer: the zeros are \( x = 1 \) and \( x = 2 \).

2. Multiplicity of Zeros

If a polynomial is fully factored as:

\[ P(x) = (x - x_1)^{m_1} (x - x_2)^{m_2} \cdots (x - x_n)^{m_n} \]

then the zero \( x = x_i \) has multiplicity \( m_i \) — the exponent of the corresponding factor. The sum of all multiplicities (real and complex zeros combined) equals the degree of the polynomial, by the Fundamental Theorem of Algebra. So the number of real zeros is at most the degree.

Example 2 — Find zeros and their multiplicities

Find the zeros and their multiplicities of \( P(x) = -2(x+1)^3(x-3)(2x+7)^2 \).

Step-by-step solution

Solve \( -2(x+1)^3(x-3)(2x+7)^2 = 0 \):

  • \( x + 1 = 0 \Rightarrow x = -1 \) — multiplicity 3
  • \( x - 3 = 0 \Rightarrow x = 3 \) — multiplicity 1
  • \( 2x + 7 = 0 \Rightarrow x = -\tfrac{7}{2} \) — multiplicity 2

Degree check: \( 3 + 1 + 2 = 6 \) ✓

3. Behavior of the Graph Near Zeros

At a real zero, the graph either crosses or touches the x-axis depending on the multiplicity:

Multiplicity Graph behavior Sign changes?
1Crosses (like a line)Yes
2Touches (like a parabola)No
3Crosses (flatter than mult 1)Yes
4Touches (flatter than mult 2)No

Example 3 — Analyzing a mixed-multiplicity graph

For \( P(x) = 0.05(x-1)^2(x-3)(x+2)^3 \), analyze the behavior at each zero.

Solution & graph
  • \( x = -2 \) — crosses (multiplicity 3, odd)
  • \( x = 1 \) — touches (multiplicity 2, even)
  • \( x = 3 \) — crosses (multiplicity 1, odd)
Graph of P(x) = 0.05(x−1)²(x−3)(x+2)³ showing a crossing at x = −2, a touch at x = 1, and a crossing at x = 3
Figure 1 — \( P(x) = 0.05(x-1)^2(x-3)(x+2)^3 \).

Example 4 — Two touching zeros and one crossing

For \( P(x) = 0.1(x-1)^2(x-3)^4(x+1)^3 \), analyze the behavior at each zero.

Solution & graph
  • \( x = -1 \) — crosses (multiplicity 3, odd)
  • \( x = 1 \) — touches (multiplicity 2, even)
  • \( x = 3 \) — touches (multiplicity 4, even)
Graph of P(x) = 0.1(x−1)²(x−3)⁴(x+1)³ showing a crossing at x = −1 and touching at x = 1 and x = 3
Figure 2 — \( P(x) = 0.1(x-1)^2(x-3)^4(x+1)^3 \).

4. Writing Polynomials from Graph Behavior

Example 5 — Construct polynomials from described behavior

(a) Find a degree-4 polynomial \( Q_1(x) \) with a zero at \( x = 2 \) (multiplicity 2) and zeros at \( x = 1 \) and \( x = -1 \).

Solution (a)

Multiplicities must sum to 4: \( 2 + 1 + 1 = 4 \) ✓.

\[ Q_1(x) = k_1(x - 2)^2(x - 1)(x + 1), \quad k_1 \ne 0 \]

(b) Find a degree-5 polynomial \( Q_2(x) \) whose graph touches the x-axis at \( x = 0 \) and \( x = 4 \), and crosses at \( x = 3 \).

Solution (b)

Touching at 0 and 4 → even multiplicity (2 each). Crossing at 3 → odd multiplicity (1). Sum: \( 2 + 2 + 1 = 5 \) ✓.

\[ Q_2(x) = k_2 x^2 (x - 4)^2 (x - 3), \quad k_2 \ne 0 \]

5. Practice Questions

Part A — Determine zeros, multiplicities, and graph behavior

(a) \( P_1(x) = 0.5(x + 1)(x - 2)(x - 3)^2 \)

Solution

Zeros and multiplicities:

  • \( x = -1 \) — multiplicity 1 (crosses)
  • \( x = 2 \) — multiplicity 1 (crosses)
  • \( x = 3 \) — multiplicity 2 (touches)

Degree check: \( 1 + 1 + 2 = 4 \) ✓

Graph of P₁(x) = 0.5(x+1)(x−2)(x−3)², crossing at x = −1 and x = 2, touching at x = 3
Figure 3 — \( P_1(x) \).

(b) \( P_2(x) = 0.1(x - 1)^3(x + 4)^2 \)

Solution

Zeros and multiplicities:

  • \( x = 1 \) — multiplicity 3 (crosses)
  • \( x = -4 \) — multiplicity 2 (touches)

Degree check: \( 3 + 2 = 5 \) ✓

Graph of P₂(x) = 0.1(x−1)³(x+4)², crossing at x = 1 and touching at x = −4
Figure 4 — \( P_2(x) \).

(c) \( P_3(x) = 0.03 x^3 (x - 3)^2 (x + 3)^2 \)

Solution

Zeros and multiplicities:

  • \( x = 0 \) — multiplicity 3 (crosses)
  • \( x = 3 \) — multiplicity 2 (touches)
  • \( x = -3 \) — multiplicity 2 (touches)

Degree check: \( 3 + 2 + 2 = 7 \) ✓

Graph of P₃(x) = 0.03x³(x−3)²(x+3)², crossing at x = 0 and touching at x = 3 and x = −3
Figure 5 — \( P_3(x) \).

Part B — Write polynomials in factored form

(a) Degree 3, zero at \( x = -1 \), graph touches \( x = -3 \)

Solution

Touches at \( x = -3 \) → even multiplicity 2. Crosses at \( x = -1 \) → multiplicity 1. Sum: \( 2 + 1 = 3 \) ✓.

\[ Q_1(x) = k_1 (x + 1)(x + 3)^2, \quad k_1 \ne 0 \]

(b) Degree 4, touches at \( x = 0 \), crosses at \( x = 3 \) and \( x = \tfrac{1}{2} \)

Solution

Touches at 0 → multiplicity 2. Crosses at 3 and \( 1/2 \) → multiplicity 1 each. Sum: \( 2 + 1 + 1 = 4 \) ✓.

\[ Q_2(x) = k_2 x^2 (x - 3)(x - \tfrac{1}{2}), \quad k_2 \ne 0 \]

(c) Degree 6, touches at \( x = -1 \), crosses at \( x = 3 \) and \( x = -4 \), with multiplicity at \( x = 3 \) > multiplicity at \( x = -4 \)

Solution

Touches at \( -1 \) → even multiplicity. Crosses at \( 3 \) and \( -4 \) → odd multiplicities. Sum must be 6, with mult(3) > mult(−4).

One valid choice: mult(−1) = 2, mult(3) = 3, mult(−4) = 1. Sum = \( 2 + 3 + 1 = 6 \) ✓.

\[ Q_3(x) = k_3 (x + 1)^2 (x - 3)^3 (x + 4), \quad k_3 \ne 0 \]

Frequently Asked Questions

What is the multiplicity of a zero?

If \(P(x) = (x - x_1)^{m_1} (x - x_2)^{m_2} \cdots\), the multiplicity of \(x = x_i\) is the exponent \(m_i\) — the number of times \((x - x_i)\) appears in the factorization.

How does multiplicity affect the graph of a polynomial?

Odd multiplicity (1, 3, 5, …) → graph crosses the x-axis. Even multiplicity (2, 4, 6, …) → graph touches the x-axis and turns around (no sign change).

What is the sum of all multiplicities?

The sum over all zeros (real + complex) equals the degree, by the Fundamental Theorem of Algebra. The number of real zeros counting multiplicity is at most the degree.

How do you write a polynomial with a given graph shape?

For each zero \(c\), include a factor \((x - c)^m\), where \(m\) is the multiplicity. Even \(m\) ⇒ touching; odd \(m\) ⇒ crossing. The multiplicities must add to the degree. Multiply by an unknown leading coefficient \(k\), determined by any other point on the graph.

Further Reading and Resources

Last updated: October 17, 2026