Polynomial Functions: Zeros, Multiplicity, Degree, and Graphs

Q4. Analyze the polynomial function graphed below:

Background

Topic: Polynomial Functions and Their Graphs

This question tests your ability to interpret a polynomial graph, identify its zeros, determine the multiplicity of each zero, and analyze the degree and leading coefficient of the function.

Key Terms and Formulas:

• Zero (Root): A value of $x$ where $f(x) = 0$.

• Multiplicity: The number of times a zero is repeated. If the graph crosses the x-axis at a zero, the multiplicity is odd; if it touches and turns around, the multiplicity is even.

• Degree: The highest power of $x$ in the polynomial.

• Leading Coefficient: The coefficient of the term with the highest degree.

Step-by-Step Guidance

1. Identify the x-values where the graph crosses or touches the x-axis. These are the zeros of $f(x)$.

2. For each zero, observe the behavior of the graph at that point:

   • If the graph crosses the axis, the zero has odd multiplicity.

   • If the graph touches and turns around, the zero has even multiplicity.

3. Count the total number of zeros and their multiplicities to estimate the degree of the polynomial.

4. Analyze the end behavior of the graph to determine if the degree is even or odd, and whether the leading coefficient is positive or negative.

Try solving on your own before revealing the answer!

Final Answer:

The zeros are $x = -8, -4, 0, 4$.

Multiplicity: $x = -8$ (even), $x = -4$ (odd), $x = 0$ (odd), $x = 4$ (even).

Degree: Even.

Leading coefficient: Positive.

The graph crosses at $x = -4$ and $x = 0$ (odd multiplicity), and touches at $x = -8$ and $x = 4$ (even multiplicity). The ends of the graph both rise, indicating an even degree and positive leading coefficient.

Q5. Analyze the polynomial function graphed below:

Background

Topic: Polynomial Functions and Their Graphs

This question tests your ability to determine the degree, leading coefficient, and zeros (with multiplicity) of a polynomial from its graph.

Key Terms and Formulas:

• Zero (Root): A value of $x$ where $f(x) = 0$.

• Multiplicity: The number of times a zero is repeated.

• Degree: The highest power of $x$ in the polynomial.

• Leading Coefficient: The coefficient of the term with the highest degree.

Step-by-Step Guidance

1. Identify the x-values where the graph crosses the x-axis. These are the zeros of $f(x)$.

2. Observe the behavior at each zero to determine multiplicity (crossing = odd, touching = even).

3. Analyze the end behavior of the graph to determine if the degree is even or odd.

4. Check if the graph opens upward or downward to determine the sign of the leading coefficient.

Try solving on your own before revealing the answer!

Final Answer:

Degree: Even.

Leading coefficient: Positive.

Zeros: $x = -6$ (multiplicity 1), $x = -4$ (multiplicity 1).

The graph crosses the x-axis at both zeros, indicating odd multiplicity. The ends of the graph rise, indicating an even degree and positive leading coefficient.

Q8. Find the formula for the polynomial function of least degree given the graph below:

Background

Topic: Constructing Polynomial Functions from Graphs

This question tests your ability to write a polynomial function based on its graph, using the zeros and their multiplicities.

Key Terms and Formulas:

• Zero (Root): A value of $x$ where $f(x) = 0$.

• Multiplicity: The number of times a zero is repeated.

• General Form: $f(x) = a(x - r_1)^{m_1}(x - r_2)^{m_2}\ldots$

Step-by-Step Guidance

1. Identify the zeros of the function from the graph (where the curve crosses or touches the x-axis).

2. Determine the multiplicity of each zero by observing the behavior at each zero (crossing = odd, touching = even).

3. Write the general form of the polynomial using the zeros and their multiplicities.

4. Estimate the leading coefficient $a$ by considering the end behavior of the graph.

Try solving on your own before revealing the answer!

Final Answer:

$f(x) = a(x + 2)(x - 1)^3$

The graph has zeros at $x = -2$ (multiplicity 1) and $x = 1$ (multiplicity 3). The end behavior suggests the sign of $a$.