Problem 1 - Precalculus: Functions and Graphs

Question

Find a polynomial \(f(x)\) of degree 3 that has the indicated zeros and satisfies the given condition. $$-1,2,3 ; \quad f(-2)=80$$

Step-by-Step Solution

Answer

The polynomial is \(f(x) = -4(x + 1)(x - 2)(x - 3)\).

Step 1: Understand the Polynomial Structure

Since the polynomial is of degree 3 and given zeros are \(-1, 2, 3\), the polynomial can be expressed as \(f(x) = a(x + 1)(x - 2)(x - 3)\). We introduced the leading coefficient \(a\) since the polynomial could be any multiple of the basic form.

Step 2: Substitute the Condition

Given \(f(-2) = 80\), substitute \(x = -2\) into the polynomial. This results in \(f(-2) = a(-2 + 1)(-2 - 2)(-2 - 3) = 80\). Simplify to get \(a(-1)(-4)(-5) = 80\).

Step 3: Simplify and Solve for the Coefficient

Calculate the expression \(-1 \times -4 \times -5 = -20\), so the equation becomes \(-20a = 80\). Solve for \(a\) by dividing both sides by \(-20\), giving \(a = -4\).

Step 4: Write the Polynomial with the Coefficient

Substitute \(a = -4\) back into the polynomial expression to get the final polynomial: \(f(x) = -4(x + 1)(x - 2)(x - 3)\).

Key Concepts

Degree of a Polynomial

The degree of a polynomial is a fundamental concept that tells us about the largest exponent of the variable in a polynomial expression. In simpler terms, it is the highest power of the variable present in the expression when it is expanded. For instance, in the polynomial expression \(f(x) = x^3 + 2x^2 + x + 1\), the degree is 3 because the highest exponent of x is 3.

• The degree determines the number of solutions or roots the polynomial can have.
• It also suggests the behavior of the polynomial graph as x approaches positive or negative infinity.

In a polynomial problem, identifying the degree helps in understanding the possible complexity and number of roots. This is particularly crucial when creating a polynomial from given zeros, as in the exercise above, which specifically mentioned a polynomial of degree 3.

Leading Coefficient

The leading coefficient of a polynomial is the coefficient of the term with the highest degree. In our exercise, after determining the possible structure of the polynomial from the given zeros, we defined the polynomial as \(f(x) = a(x + 1)(x - 2)(x - 3)\). Here, the variable 'a' represents the leading coefficient when the polynomial is fully expanded.

• The leading coefficient affects the steepness and direction of the graph of the polynomial.
• For example, a positive leading coefficient means the graph will eventually rise as it moves further away from the origin, while a negative one means it will fall.

In the solution, we found that the leading coefficient 'a' is -4, which is crucial for satisfying the condition \(f(-2) = 80\). By solving for 'a', we tailor the polynomial precisely to meet all given conditions.

Zeros of a Polynomial

Zeros of a polynomial, also known as roots, are the solutions to the equation \(f(x) = 0\). They are the x-values where the polynomial meets the x-axis. In our specific exercise, the zeros provided were -1, 2, and 3. These zeros can be found by setting each factor of the polynomial to zero:

• Zero at -1 from the factor \(x + 1\)
• Zero at 2 from the factor \(x - 2\)
• Zero at 3 from the factor \(x - 3\)

Identifying zeros is critical when building a polynomial because it determines the fundamental layout of the polynomial equation. The zeros directly influence the roots which, in turn, determine the overall graph structure and solutions.

Polynomial Roots

Polynomial roots are synonymous with the zeros of the polynomial - they are the values of \(x\) which satisfy \(f(x) = 0\). Finding the roots of a polynomial is crucial as they help in graphing the function and understanding its behavior. In our exercise, we used the given roots of -1, 2, and 3 to establish the foundational structure of the polynomial function.

• Each root corresponds to a factor of the polynomial.
• The process of finding a polynomial from known roots involves reversing polynomial division.

In practical scenarios, the roots of a polynomial give us insights into its graph - where it crosses or touches the x-axis. Positive, negative, or zero roots can alter the layout and direction of the polynomial graph significantly.