Problem 22 - Precalculus with Calculus Previews

Question

Construct a polynomial function \(f\) that has the given properties. There is no unique answer. \(f\) is of degree 6 , has four real zeros, 2 is a zero of multiplicity 3 , behaves like \(y=2 x^{6}\) for large values of \(|x|, f(0)=8\)

Step-by-Step Solution

Answer

The polynomial is \( f(x) = 2(x-2)^3\left(x+\frac{1}{\sqrt[3]{2}}\right)^3 \).

Step 1: Understand the Polynomial Degree

Identify that the polynomial has a degree of 6, implying it will have a total of 6 roots, counting multiplicities.

Step 2: Identify Zero and its Multiplicity

Since 2 is a zero of multiplicity 3, we express part of the polynomial as \( (x-2)^3 \). This accounts for three out of six roots.

Step 3: Determine Additional Real Zeros

Since there are four real zeros in total, we need one more distinct real root to check the total multiplicity. Assume \( a \) is a real zero with multiplicity 3, making the polynomial \( (x-2)^3(x-a)^3 \).

Step 4: Evaluate Leading Coefficient

The function behaves like \( y = 2x^6 \), so the leading coefficient of the polynomial should be 2. Thus, the polynomial is of the form \( f(x) = 2(x-2)^3(x-a)^3 \).

Step 5: Set the y-Intercept

Substitute \( x = 0 \) into the polynomial and set it equal to 8, as \( f(0) = 8 \). This gives the equation \( 2(-2)^3(-a)^3 = 8 \).

Step 6: Solve for the Unknown Zero

First compute \( 2(-2)^3 = -16 \). Substitute into the equation: \( -16(-a)^3 = 8 \), giving \( (-a)^3 = -\frac{1}{2} \). Thus \( a = -\frac{1}{\sqrt[3]{2}} \).

Step 7: Write the Polynomial Function

Substitute back the value for \( a \) into the polynomial. The function is \( f(x) = 2(x-2)^3\left(x+\frac{1}{\sqrt[3]{2}}\right)^3 \).

Key Concepts

Degree of Polynomial

The degree of a polynomial is an essential concept, as it determines the polynomial's behavior and characteristics. In this exercise, we have a polynomial of degree 6. This means it will have a total of 6 roots, which include all multiplicities. The degree of a polynomial also affects its shape and the number of turning points. For a polynomial of degree 6, the graph may have up to 5 turning points.

When creating a polynomial with a specific degree, it's important to ensure the sum of root multiplicities equals the degree. Here, since we need 6 roots, our task is to assign and account for these roots correctly across real zeros, with one of these zeros having a specified multiplicity.

Real Zeros

Real zeros are the points where the polynomial crosses or touches the x-axis. They are the solutions to the polynomial equation when set to zero. In this exercise, we are required to have four real zeros, which are the actual values of x for which the function equals zero.

The real zeros provide critical information about how the polynomial behaves and how it appears graphically. However, because zero multiplicity is involved, some of these zeros may not appear as individual distinct intersections on the graph. Instead, they might simply touch the x-axis without crossing.

Understanding where these zeros are located helps in sketching the graph and predicting the function's behavior. Knowing that the polynomial has real zeros guides us in constructing the function accurately.

Zero Multiplicity

When discussing polynomial zeros, multiplicity refers to the number of times a particular zero appears in the polynomial. In this case, the zero 2 has a multiplicity of 3, meaning the factor \( (x-2)^3 \) appears in the polynomial.

Zero multiplicity affects how the graph behaves at that zero. For instance, a zero with an odd multiplicity, like 3, will cause the graph to cross the x-axis at that point. Conversely, an even multiplicity will cause the graph to merely touch the x-axis and turn around. Multiplicity is crucial for determining the exact form and graphical representation of polynomial functions.

Aside from its impact on graph shape, zero multiplicity is important in resolving the overall structure and that measurement equates perfectly with the initial degree set for the polynomial.

Leading Coefficient

The leading coefficient is the coefficient of the term with the highest degree in a polynomial. It plays a key role in determining the end behavior of the polynomial's graph. For this exercise, the leading coefficient is specified as 2, indicating that as \(|x|\) becomes very large, the polynomial behaves like \( y = 2x^6 \).

The leading coefficient influences both the width and direction of the end behavior of the graph. Positive leading coefficients indicate that the graph rises on both ends if the degree is even, and negative if the degree is odd.

• This means in our case with a degree of 6, the graph will rise on both ends because the leading coefficient is positive.

The leading coefficient also directly affects the polynomial’s critical points and must be considered carefully to respect the polynomial's specified traits.