Real Zeros and Graphs of Polynomials

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The effects of the zeros of a polynomial on its graph are investigated using examples and questions and their solutions.
Although we use polynomials in factored forms, factoring polynomials is also included.
More advanced topics on finding zeros of polynomials and also an interactive app to investigate graphs of polynomial functions are included.

Zeros of a Polynomial

Let \( P(x) \) be a polynomial. The zeros of a polynomial are all values of \( x \) that are solutions to the equation \( P(x) = 0 \). The real zeros of a polynomial correspond to the \( x \) - intercepts of the graph of the polynomial. The graph of a polynomial either touches or crosses the \( x \) - axis at the zeros. Before finding the zeros of a polynomial, we first need to factor it and it is included if needed.

Example 1
Find the zeros of the polynomial \( P(x) = 5 (x - 1) (x - 2) \).
Solution
The zeros of polynomial \( P \) are found by solving the equation: \( \quad 5 (x - 1) (x - 2) = 0 \)
\( x - 1\) and \( x -2 \) are fators (dependent on \( x \) ) and therefore \( P(x) = 0 \) if any of the two factors in \( x \) is equal to zero. Hence
\( x - 1 = 0 \) or \( x - 2 = 0 \)
The solutions of the above linear equations are: \( x = 1 \) and \( x = 2 \)
The zeros of \( P(x) \) are \( x = 1 \) and \( x = 2 \).

Multiplicity of Zeros of Polynomial

When a polynomial \( P(x) \) is completely factored over the real numbers as follows: \( P(x) = (x - x_1)^{m_1} (x - x_2)^{m_2} ... (x - x_n)^{m_n} \), then the zero \( x = x_1 \) has a multiplicity \( m_1 \), the zero \( x = x_2 \) has a multiplicity \( m_2 \), etc ...
Therefore the multiplicity of a zero of a polynomial is the number of times the factor associated with that zero appears in the factored form of the polynomial. This is shown in the exponent.
Note that zeros of a given polynomial are in general complex numbers. The total number of zeros is equal to the degree of the polynomial. Hence the total number of real zeros is less than or equal to the degree of the given polynomial.

Example 2
Find the zeros and their multiplicities of the polynomial: \( \quad P(x) = -2 (x + 1)^3 (x - 3) (2x + 7)^2 \).
Solution
The zeros of polynomial \( P \) are found by solving the equation: \( \quad -2 (x + 1)^3 (x - 3) (2x + 7)^2 = 0 \)
\( x + 1 \), \( x - 3 \) and \( 2x + 7 \) are factors (dependent on \( x \) ) and therefore \( P(x) = 0 \) if any of the two factors is equal to zero. Hence
\( x + 1 = 0 \) or \( x - 3 = 0 \) or \( 2x + 7 = 0 \)
The solutions of the above linear equations are: \( x = -1 \) , \( x = 3 \) and \( x = - \dfrac{7}{2} \)
The zeros of \( P(x) \) and their multiplicities are:
\( x = -1 \) with multiplicity \( 3 \)
\( x = 3 \) with multiplicity \( 1 \)
\( x = - \dfrac{7}{2} \) with multiplicity \( 2 \)

behavior of the Graph of a Polynomial Near its Zeros

At a given real zero, the polynomial is equal to zero and therefore its graph crosses or touches the \( x \) axis. The zero correspond to an x - intercept in the graph of the polynomial.
The graph of a polynomial crosses the \( x \) axis at a given zero if zero has an odd multiplicity.
The graph of a polynomial touches the \( x \) axis at a given zero if zero has an even multiplicity.

Example 3
What is the behavior of the graph at the zeros of of the polynomial \( P(x) = 0.05 (x-1)^2 (x - 3) (x + 2)^3 \).
Solution
At the zero \( x = - 2 \), the graph crosses the \( x \) axis because the multiplicity of zero is equal to \( 3 \) and therefore odd.
At the zero \( x = 1 \), the graph touches the \( x \) axis because the multiplicity of zero is equal to \( 2 \) and therefore even.
At the zero \( x = 3 \), the graph crosses the \( x \) axis because the multiplicity of zero is equal to \( 1 \) and therefore odd.
The graph of the given polynomial is shown in figure 1 below and shows the described behavior above.

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

Example 4
What is the behavior of the graph at the zeros of of the polynomial \( P(x) = 0.1 (x-1)^2 (x-3)^4 (x+1)^3 \)
Solution
At the zero \( x = - 1 \), the graph crosses the \( x \) axis because the multiplicity of zero is equal to \( 3 \) and therefore odd.
At the zero \( x = 1 \), the graph touches the \( x \) axis because the multiplicity of zero is equal to \( 2 \) and therefore even.
At the zero \( x = 3 \), the graph touches the \( x \) axis because the multiplicity of zero is equal to \( 4 \) and therefore even
The graph of the given polynomial is shown in figure 2 below and shows the described behavior above.

Write Polynomials Given their Degree and behavior of the Graphs at the Zeros

Example 5
a) Write a polynomial \( Q_1(x) \) in factored form whose degree is equal to \( 4 \) has a zero at \( x = 2 \) with multiplicity \( 2 \), and zeros at \( x = 1 \) and \( x = - 1 \).
b) Write a polynomial \( Q_2(x) \) in factored form whose degree is equal to \( 5 \) and its graph touches the \( x \) axis at \( x = 0 \) and crosses the \( x \) axis at \( x = 3 \) and touches the \( x \) axis at \( x = 4 \).
Solution
a)
The zero at \( x = 2 \) has multiplicity \( 2 \), hence \( (x - 2)^2 \) is a factor of \( Q_1(x) \). The zeros at \( x = 1 \) and \( x = - 1 \) cannot have a multiplicity greater than \( 1 \) since the addition of the multiplicities of all zeros of a polynomial cannot be greater than its degree. Hence,
\( Q_1(x) = k_1 (x - 2)^2 (x-1) (x+1) \) , where \( k_1 \) is any real constant not equal to zero.
b)
The graph of \( Q_2(x) \) touches the \( x \) axis at \( x = 0 \) and \( x = 4 \) and therefore the multiplicity of the zeros at \( x = 0 \) and \( x = 4 \) must be even and cannot be greater than \( 2 \). The graph of \( Q_2(x) \) crosses the \( x \) axis at \( x = 3 \) and therefore the multiplicity of the zeros at \( x = 3 \) must be odd. Taking into account the degree \( 5 \), \( Q_2(x) \) may be written in the form
\( Q_2(x) = k_2 x^2 (x-4)^2 (x - 3) \), where \( k_2 \) is any real constant not equal to zero.

Questions

(with solutions)

Part A
For each polynomial below, determine the zeros and their multiplicities and describe the behavior of the graph of the polynomial near each zero. Use a graphing calculator to check your answers.
a) \( P_1(x) = 0.5(x+1) (x-2) (x -3)^2 \)
b) \( P_2(x) = 0.1 (x-1)^3 (x+4)^2 \)
c) \( P_3(x) = 0.03 x^3 (x-3)^2 (x+3)^2 \)

Part B
Write polynomials, in factored forms, with the following properties

a) \( Q_1(x) \) has a degree is equal to \( 3 \), has a zero at \( x = -1 \) and its graph touches the \( x \) axis at \( x = - 3 \).
b) \( Q_2(x) \) has degree is equal to \( 4 \) and its graph touches the \( x \) axis at \( x = 0 \) and crosses the \( x \) axis at \( x = 3 \) and at \( x = \dfrac{1}{2} \).
c) \( Q_3(x) \) has degree is equal to \( 6 \) and its graph touches the \( x \) axis at \( x = -1 \) and crosses the \( x \) axis at \( x = 3 \) and at \( x = - 4 \) with the multiplicity of the zero at \( x = 3 \) greater than the multiplicity of the zero at \( x = - 4 \).

Solutions to the Above Questions

Part A
a)
zero at \( x = - 1 \) with multiplicity \( 1 \).
zero at \( x = 2 \) with multiplicity \( 1 \).
zero at \( x = 3 \) with multiplicity \( 2 \).
The graph of the polynomial \( P_1(x) \) crosses the \( x \) axis at \( x = -1 \) and \( x = 2 \) and touches the \( x \) axis at \( x = 3 \).

Zeros and x intercepts of polynomials P_1 — Fig.3 - Zeros and \( x \) Intercepts of Polynomial \( P_1(x) \)

b)
zero at \( x = 1 \) with multiplicity \( 3 \).
zero at \( x = - 4 \) with multiplicity \( 2 \).
The graph of the polynomial \( P_2(x) \) crosses the \( x \) axis at \( x = 1 \) and touches the \( x \) axis at \( x = -4 \)

Zeros and x intercepts of polynomials P_2 — Fig.4 - Zeros and \( x \) Intercepts of Polynomial \( P_2(x) \)

c)
zero at \( x = 0 \) with multiplicity \( 3 \).
zero at \( x = 3 \) with multiplicity \( 2 \).
zero at \( x = - 3 \) with multiplicity \( 2 \).
The graph of the polynomial \( P_3(x) \) crosses the \( x \) axis at the origin ( \( x = 0 \) ) and touches the \( x \) axis at \( x = -3 \) and \( x = 3 \)

Zeros and x intercepts of polynomials P_3 — Fig.5 - Zeros and \( x \) Intercepts of Polynomial \( P_3(x) \)

Part B
a) \( Q_1 (x) = k_1 (x + 1) (x + 3)^2 \)
b) \( Q_2 (x) = k_2 x^2 (x - 3) (x - \dfrac{1}{2} ) \)
c) \( Q_3 (x) = k_2 (x + 1)^2 (x - 3)^3 (x + 4 ) \)