Problem 64 - College Algebra with Corequisite Support

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 \(x=2\) and a root of multiplicity 1 at \(x=-2\). \(y\) -intercept at (0,4) .

Step-by-Step Solution

Answer

The polynomial equation is \( P(x) = \frac{1}{18}(x+3)^2(x-2)^2(x+2) \).

Step 1: Identify the Root Factors

Given the roots of multiplicity, we can form the polynomial's factors. The roots are -3 (multiplicity 2), 2 (multiplicity 2), and -2 (multiplicity 1). The factors are \((x+3)^2\), \((x-2)^2\), and \((x+2)\)."

Step 2: Write the General Polynomial Equation

By multiplying the identified factors, we get the general form of the polynomial:\[ P(x) = a(x+3)^2(x-2)^2(x+2) \]where \(a\) is a constant that needs to be determined using additional information given in the problem.

Step 3: Use the y-intercept to Find 'a'

We know the polynomial passes through the point (0, 4), making it the \(y\)-intercept. Substitute \(x = 0\) and \(P(x) = 4\):\[ 4 = a(0+3)^2(0-2)^2(0+2) \]\[ 4 = a(9)(4)(2) \]\[ 4 = 72a \]Solve for \(a\):\[ a = \frac{4}{72} = \frac{1}{18} \]

Step 4: Write the Final Polynomial Equation

Substitute the value of \(a\) back into the polynomial equation:\[ P(x) = \frac{1}{18}(x+3)^2(x-2)^2(x+2) \]This is the final polynomial equation with the given roots and y-intercept.

Key Concepts

Roots and Multiplicity

In the world of polynomial equations, roots are special numbers that make the polynomial equal to zero when substituted for the variable. Each root can have a certain multiplicity, which tells us how many times that particular root appears in the factorization of the polynomial.

For instance, a root with multiplicity 2 means it appears twice, making its factor squared in the polynomial equation. If we consider the given polynomial, we have roots at \(x = -3\) and \(x = 2\) with a multiplicity of 2 each. This means the factors \((x + 3)\) and \((x - 2)\) are squared:

• Root \(-3\) with multiplicity 2: \((x + 3)^2\)
• Root \(2\) with multiplicity 2: \((x - 2)^2\)
• Root \(-2\) with multiplicity 1: \((x + 2)\)

Multiplicity affects the shape of the graph. A root of even multiplicity, like 2, means the graph "touches" the x-axis and turns around, while a root with odd multiplicity, like 1, "crosses" the x-axis. Understanding this nuance helps us predict how the graph behaves around each root.

Y-intercept

The y-intercept is a crucial part of understanding the behavior of a polynomial graph. It represents the point where the graph crosses the y-axis, meaning it is the point where \(x\) is equal to zero.

For any polynomial function \(P(x)\), finding the y-intercept involves simply setting \(x = 0\). In our example, the point given is (0, 4), meaning that when \(x = 0\), the value of the polynomial is 4. Using this information, we can compute the constant "a" that scales the polynomial to this y-intercept.

By setting \[ 4 = a(0+3)^2(0-2)^2(0+2), \]you can solve for "a" and find that \(a = \frac{1}{18}\).

This step ensures that the polynomial equation accurately represents the specific graph by confirming it passes through the correct y-intercept, crucial for graph's vertical alignment.

Degree of Polynomial

A polynomial's degree is vital as it determines the number of roots and the behavior of its graph. The degree is the highest power of the variable in the polynomial equation when it is expanded and fully simplified.

For the polynomial described in the exercise, the degree is 5. This is derived from the multiplication of its factors and terms, accounting for the multiplicity of each root: \((x+3)^2(x-2)^2(x+2)\).

To find this degree, consider:

• Factor \((x+3)^2\) contributes 2 to the degree.
• Factor \((x-2)^2\) also adds 2.
• Factor \((x+2)\) adds 1.

Adding these, 2 + 2 + 1 equals a degree of 5.

The degree dictates not only the number of roots but also the end behavior of the polynomial. Higher degree polynomials have more complex shapes and crossing points. Thus, knowing the degree is important for understanding the overall form of the graph.