Problem 67 - College Algebra with Corequisite Support

Question

For the following exercises, construct a polynomial function of least degree possible using the given information. Real roots: -1,1 (with multiplicity 2 and 1\()\) and \((2, f(2))=(2,4)\)

Step-by-Step Solution

Answer

The polynomial is \(f(x) = \frac{4}{9}(x + 1)^2(x - 1)\).

Step 1: Identify roots and their multiplicity

The polynomial function will include the given real roots. The roots are -1 with multiplicity 2 and 1 with multiplicity 1. Thus, the factors associated with these roots are \((x + 1)^2\) and \((x - 1)\).

Step 2: Introduce the unknown factor for degree adjustment

Because the polynomial must pass through the point \((2, 4)\), we introduce a constant factor \(a\) to the polynomial to adjust its degree and ensure it fits all conditions. Thus the polynomial can be expressed as\(f(x) = a(x + 1)^2(x - 1)\).

Step 3: Use the point (2,4) to find a

Substitute \(x = 2\) and \(f(2) = 4\) into the polynomial \(f(x) = a(x + 1)^2(x - 1)\) to find \(a\). Solve:\[f(2) = a(2 + 1)^2(2 - 1) = 4\]\[f(2) = a(3)^2(1) = 4\]\[a \cdot 9 = 4\] \[a = \frac{4}{9}\].

Step 4: Construct the polynomial function

Now that we've calculated \(a\), the polynomial function is:\[ f(x) = \frac{4}{9}(x + 1)^2(x - 1) \]

Key Concepts

Real Roots

In the world of polynomial functions, real roots are the values of \(x\) for which the polynomial equals zero. For the given problem, we have identified the real roots as \(-1\) and \(1\). These correspond to the points where the graph of the polynomial intersects the x-axis. The involvement of real roots is crucial, as they form the basis for our polynomial equation construction.

The root \(-1\) has a specific nature, described by ‘multiplicity’. We'll dive deeper into that concept next. The root \(1\), however, occurs just once. Understanding the repeated nature or singular nature of each root allows us to predict and construct polynomial expressions effectively.

Multiplicity of Roots

Multiplicity refers to how many times a particular root repeats within a polynomial. If a root appears more than once, its effect on the graph is distinct. In our exercise, the root \(-1\) has a multiplicity of 2, while root \(1\) has a multiplicity of 1.

• If a root has an odd multiplicity, like \(1\), the graph will cross the x-axis at this point.
• If a root has an even multiplicity, like \(-1\) with multiplicity 2, the graph will "touch" the x-axis and bounce back without crossing it.

These aspects greatly influence the shape and behavior of the graph. Recognizing the multiplicity allows you to write each root factor properly, using exponents to denote their multiplicity, such as \((x + 1)^2\) for \(-1\), and \((x - 1)\) for \(1\).

Factoring Polynomials

Factoring a polynomial means expressing it as a product of its factors. It’s like breaking down a complex expression into simpler pieces. For the polynomial function in our exercise, we start with the roots to form the factors:
\[(x + 1)^2\] for the root \(-1\), and \[(x - 1)\] for the root \(1\).

Combining these factors gives us the basic structure of the polynomial. But to account for real-world conditions, such as passing through a specific point, we may introduce constants. This condition is met by multiplying these factors by a constant \(a\), a necessary tweak to fit the polynomial to given constraints.

Point Evaluation

Point evaluation is a technique to find the exact value of a polynomial at a given \(x\). In our problem, we were tasked with ensuring that the polynomial satisfies the condition \((2, f(2)) = (2, 4)\).

This is achieved by substituting \(x = 2\) into the polynomial \(f(x) = a(x + 1)^2(x - 1)\). Solving for \(a\), through:
\[4 = a(3)^2(1)\]
helps you derive \(a = \frac{4}{9}\). This constant \(a\) ensures the polynomial curves correctly to pass through the given point, adjusting the shape to match precise data conditions. It seamlessly connects all the equations criteria with real data needs, providing a solution that not only meets algebraic requirements but real-world accuracy.