Problem 66 - College Algebra with Corequisite Support

Question

For the following exercises, construct a polynomial function of least degree possible using the given information. Real roots: -1,1,3 and \((2, f(2))=(2,4)\)

Step-by-Step Solution

Answer

The polynomial function is \( f(x) = -\frac{4}{3}(x^3 - 3x^2 - x + 3) \).

Step 1: Identify the Roots

We are given that the function has real roots at -1, 1, and 3. This means the polynomial function will have factors of \((x + 1), (x - 1), \text{ and } (x - 3)\).

Step 2: Construct the Basic Polynomial

Using the identified roots, construct the basic form of the polynomial function:\( f(x) = a(x + 1)(x - 1)(x - 3) \).

Step 3: Simplify the Polynomial

Expand the basic polynomial form:\((x + 1)(x - 1) = x^2 - 1\).Next, use this result to multiply with \((x - 3)\):\((x^2 - 1)(x - 3) = x^3 - 3x^2 - x + 3 \).Thus, the polynomial becomes \( f(x) = a(x^3 - 3x^2 - x + 3) \).

Step 4: Use Given Point to Determine a

The problem states that \((2, f(2)) = (2, 4)\). Substitute \(x = 2\) and \(f(x) = 4\) into the polynomial:\[ 4 = a((2)^3 - 3(2)^2 - 2 + 3) \].This simplifies to \[ 4 = a(8 - 12 - 2 + 3) \].Compute:\[ 4 = a(-3) \] which simplifies to \[ a = -\frac{4}{3} \].

Step 5: Write the Final Polynomial Function

With the value of \(a\) found, the polynomial function is:\[ f(x) = -\frac{4}{3}(x^3 - 3x^2 - x + 3) \].

Key Concepts

Roots of Polynomials

The roots of a polynomial, often referred to as its zeros, are the values of the variable that make the polynomial equal to zero. When constructing a polynomial based on given roots, each root represents a factor of the polynomial. For instance, if a polynomial has roots at \(-1, 1,\) and \(3\), it means:

• \(x + 1\) is a factor because \(x = -1\) is a root.
• \(x - 1\) is a factor because \(x = 1\) is a root.
• \(x - 3\) is a factor because \(x = 3\) is a root.

These factors form the basic structure of the polynomial. To build the polynomial, multiply these factors together.

Polynomial Construction

Constructing a polynomial involves identifying its factors based on known roots and combining them. Once the roots are established, they can be expressed as linear factors. The polynomial is then formed by multiplying these factors together. In our example, combining the factors \((x + 1)\), \((x - 1)\), and \((x - 3)\) results in the polynomial function:\[ f(x) = a(x + 1)(x - 1)(x - 3) \]where \(a\) is a constant multiplier used to adjust the polynomial for specific conditions, like passing through a given point. Once "a" is found, substitute it back to complete the function.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations. In polynomial expressions, terms are usually organized from the highest to the lowest degree of the variable. Simplification involves combining like terms and following operations like distribution exemplified by expanding \((x + 1)(x - 1) = x^2 - 1\)for just two factors. In our example, the multiplication of factors expands to:\((x^2 - 1)(x - 3) = x^3 - 3x^2 - x + 3\)Each term is carefully calculated, ensuring proper distribution of terms to maintain correctness.

Function Evaluation

Function evaluation is the process of determining the output of a function for a specific input. With polynomial functions, substitute the given value into the function and solve for the output. In the exercise, \(f(2) = 4\)is used to solve for the unknown constant "a". Insert \(x = 2\) into the polynomial:\[4 = a(2^3 - 3(2)^2 - 2 + 3)\]Simplifying the terms comes to:\[4 = a(8 - 12 - 2 + 3)\]Finally, solving for \(a\) helps ensure the function passes through the given point, refining the polynomial to meet specific conditions.