Real Zeros, Multiplicity, and Graphs of Polynomial Functions

This tutorial explains how the real zeros of a polynomial affect its graph. Examples, exercises, and solutions are included. Although we focus on polynomials in factored form, factoring polynomials is also discussed. For more advanced topics, see finding zeros of polynomials and interactive polynomial graphs.

1. Zeros of a Polynomial

Let \(P(x)\) be a polynomial. The zeros are all values of \(x\) for which \(P(x) = 0\). The real zeros correspond to the \(x\)-intercepts of the graph, where the polynomial either crosses or touches the \(x\)-axis.

Example 1

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

Solution:

Solve \(5(x-1)(x-2) = 0\). The factors give: \[ x - 1 = 0 \quad \text{or} \quad x - 2 = 0 \] Hence, the zeros are: \[ x = 1, \quad x = 2 \]

2. Multiplicity of Zeros

If a polynomial is factored as \[ P(x) = (x - x_1)^{m_1} (x - x_2)^{m_2} \dots (x - x_n)^{m_n}, \] then the zero \(x=x_i\) has multiplicity \(m_i\). Multiplicity indicates how many times a zero occurs in the factorization. The total number of zeros (including complex) equals the degree of the polynomial, so the number of real zeros is at most equal to the degree.

Example 2

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

Solution:

Solve \(-2 (x+1)^3 (x-3) (2x+7)^2 = 0\). \[ x+1=0 \Rightarrow x=-1 \text{ (multiplicity 3)}, \quad x-3=0 \Rightarrow x=3 \text{ (multiplicity 1)}, \quad 2x+7=0 \Rightarrow x=-\frac{7}{2} \text{ (multiplicity 2)} \]

3. Behavior of Graph Near Zeros

At a real zero, the graph either crosses or touches the \(x\)-axis:

Example 3

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

Solution:

Zeros and x intercepts of polynomials
Fig.1 - Multiplicity of Zeros and \(x\)-Intercepts

Example 4

For \(P(x) = 0.1 (x-1)^2 (x-3)^4 (x+1)^3\):

Solution:

Zeros and x intercepts of polynomials
Fig.2 - Multiplicity of Zeros and \(x\)-Intercepts

4. Writing Polynomials from Graph Behavior

Example 5

a) Polynomial \(Q_1(x)\) of degree 4 with zero \(x=2\) (multiplicity 2) and zeros \(x=1, -1\): \[ Q_1(x) = k_1 (x-2)^2 (x-1)(x+1), \quad k_1 \neq 0 \]

b) Polynomial \(Q_2(x)\) of degree 5, graph touches at \(x=0,4\) and crosses at \(x=3\): \[ Q_2(x) = k_2 x^2 (x-4)^2 (x-3), \quad k_2 \neq 0 \]

5. Questions (with solutions)

Part A

Determine zeros, multiplicities, and behavior of graphs.

  1. \(P_1(x) = 0.5(x+1)(x-2)(x-3)^2\)
  2. \(P_2(x) = 0.1(x-1)^3 (x+4)^2\)
  3. \(P_3(x) = 0.03 x^3 (x-3)^2 (x+3)^2\)

Part B

Write polynomials in factored form:

  1. Degree 3, zero at \(x=-1\), graph touches \(x=-3\)
  2. Degree 4, touches at \(x=0\), crosses at \(x=3\), crosses at \(x=\frac12\)
  3. Degree 6, touches at \(x=-1\), crosses at \(x=3\) and \(x=-4\), multiplicity at \(x=3\) > multiplicity at \(x=-4\)

Solutions

Part A

  1. \(P_1(x)\): zeros: \(x=-1\) (1), \(x=2\) (1), \(x=3\) (2); crosses at -1,2, touches at 3
    P1 polynomial graph
    Fig.3 - Zeros and \(x\)-Intercepts of \(P_1(x)\)
  2. \(P_2(x)\): zeros: \(x=1\) (3), \(x=-4\) (2); crosses at 1, touches at -4
    P2 polynomial graph
    Fig.4 - Zeros and \(x\)-Intercepts of \(P_2(x)\)
  3. \(P_3(x)\): zeros: \(x=0\) (3), \(x=3\) (2), \(x=-3\) (2); crosses at 0, touches at ±3
    P3 polynomial graph
    Fig.5 - Zeros and \(x\)-Intercepts of \(P_3(x)\)

Part B

  1. \(Q_1(x) = k_1 (x+1)(x+3)^2\)
  2. \(Q_2(x) = k_2 x^2 (x-3)(x-\frac12)\)
  3. \(Q_3(x) = k_3 (x+1)^2 (x-3)^3 (x+4)\)

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