Problem 281 - Precalculus

Question

For the following exercises, use the given information about the polynomial graph to write the equation. Degree \(5 .\) Roots of multiplicity 2 at \(x=3\) and \(x=1,\) and a root of multiplicity 1 at \(x=-3 .\) y-intercept at \((0,9)\)

Step-by-Step Solution

Answer

The polynomial equation is \(f(x) = (x-3)^2(x-1)^2(x+3)\).

Step 1: Identify the Polynomial Roots

The roots provided are: \(x = 3\) with multiplicity 2, \(x = 1\) with multiplicity 2, and \(x = -3\) with multiplicity 1. The polynomial can be represented as a product of factors corresponding to these roots.

Step 2: Construct the Polynomial Equation

The polynomial can be expressed using the roots: \(f(x) = a(x-3)^2(x-1)^2(x+3)\). Here, \(a\) is a constant to be determined.

Step 3: Substitute the Y-Intercept

We know that the y-intercept of the polynomial is \((0,9)\), hence \(f(0) = 9\). Substituting \(x = 0\) into the equation we get \(f(0) = a(0-3)^2(0-1)^2(0+3) = 9\).

Step 4: Solve for the Constant \(a\)

Substituting into the equation, we have \(9a = 9\), which simplifies to \(a = 1\).

Step 5: Write the Final Equation

The polynomial equation is \(f(x) = (x-3)^2(x-1)^2(x+3)\) after substituting \(a = 1\). The polynomial correctly represents the roots and the y-intercept.

Key Concepts

Roots of Polynomials

Understanding roots of polynomials means knowing where the polynomial expression equals zero. Each root represents a solution to the equation formed by setting the polynomial equal to zero. In simpler terms, if you were to graph the polynomial on a coordinate plane, the roots would be the points where the curve crosses the x-axis. These points on the x-axis are where the output of the polynomial function is zero.

For instance, given the roots in the example problem as \(x = 3\), \(x = 1\), and \(x = -3\), each of these values for \(x\) makes the polynomial equal zero. If you insert any of these root values into the polynomial equation \(f(x)\), the result will be zero.

• This feature is fundamental in shaping the polynomial's graph.
• The specific behavior of each root is dictated by its multiplicity.

Multiplicity of Roots

The multiplicity of polynomial roots indicates how many times a particular root is repeated in the equation. Think of it as a count of the root's "weight" or influence on the graph's shape.

In the provided problem, the roots \(x = 3\) and \(x = 1\) both have a multiplicity of 2, meaning each root is repeated twice in the polynomial equation. This typically causes the graph to just touch the x-axis at these roots without crossing it. The root \(x = -3\) has a multiplicity of 1, creating a simple crossing of the x-axis.

• Multiplicity 1: Graph crosses the x-axis.
• Multiplicity more than 1: Graph may touch but typically doesn't cross the axis at that point.

This concept helps predict how a polynomial graph behaves around its roots, making polynomial analysis more accessible.

Y-Intercept

In a polynomial function, the y-intercept is the point where the graph crosses the y-axis. This gives valuable insight into how the polynomial behaves at \(x = 0\). To find the y-intercept, substitute \(x = 0\) in the polynomial equation and solve for \(f(x)\).

For our example, the polynomial equation is \(f(x) = (x-3)^2(x-1)^2(x+3)\). By substituting \(x = 0\) into this equation, you determine that the y-intercept occurs at \((0, 9)\), meaning the y-coordinate of the graph where \(x\) is zero is 9.

• Provides another check for the graph's accuracy.
• Shows where the graph crosses the y-axis.

Degree of Polynomial

The degree of a polynomial tells us about the highest power of \(x\) in its expression. It gives critical information about the polynomial's overall shape and behavior. In general, a polynomial of degree \(n\) can have up to \(n\) real roots and can also potentially have up to \(n-1\) turning points.

For the polynomial in our problem, it has a degree of 5. This results from adding up the multiplicities of its roots: two roots of multiplicity 2 and one root of multiplicity 1, giving a total degree of \(2 + 2 + 1 = 5\). The degree indicates that the polynomial graph is more complex, showcasing a sweeping curve with several up and down movements. It helps anticipate the general layout of the polynomial's graph even before plotting.

• Informs the polynomial structure.
• Predicts the number of x-intercepts and possible turning points.

Understanding the degree is crucial in grasping the larger picture of a polynomial function.