Problem 22 - Precalculus : Building Concepts and Connections

Question

Use synthetic division to find the function values. \(f(x)=-2 x^{3}+4 x^{2}-7 ;\) find \(f(4)\) and \(f(-3)\)

Step-by-Step Solution

Answer

The value of \(f(4)\) is 5 and the value of \(f(-3)\) is 20.

Step 1: Understanding Synthetic Division

Synthetic division is a short-cut method of performing long division in polynomial functions. It only works when you are dividing by a linear binomial (a polynomial of degree 1). Here, we will use synthetic division to find the values of the function for given x-values.

Step 2: Evaluate f(4)

To evaluate \(f(4)\), set up the operation of synthetic division using the coefficients of the polynomial and the number 4. The coefficients of the polynomial \( -2, 4, -7 \) are written in a row. Write '4' to the left. Write down the first coefficient -2, and multiply '4' by this number, add the result to the next coefficient, and continue this process. The last number you get is the result of the function.

Step 3: Evaluate f(-3)

Repeat the process from step 2, but now use -3 instead of 4. Write down the first coefficient, and multiply '-3' by this number, add the result to the next coefficient, and continue this process. The last number you get is the result of the function.

Key Concepts

Polynomial Functions

Polynomial functions are fundamental mathematical expressions that consist of variables raised to various exponents and multiplied by coefficients. They can have one or more terms and exhibit different degrees depending on the highest power of the variable they contain. A polynomial of degree 3, like the one given in the exercise (\(-2x^3 + 4x^2 - 7\)), is called a cubic polynomial.

These functions are important as they create smooth, continuous curves and appear frequently in algebra and calculus. Polynomials are easy to work with and their calculations can provide insights into roots, intercepts, and turning points, translating real-world scenarios into mathematical models. In this exercise, the polynomial function is evaluated for specific input values using a method known as synthetic division.

Function Evaluation

Evaluating a function at a given point simply means substituting the specified value of the variable into the function, and then calculating the result. For polynomial functions, this will give you the output or 'y-value' that corresponds to a certain 'x-value'.

When solving for \(f(4)\) and \(f(-3)\) using the given polynomial, this means finding out what the value of the expression \(-2x^3 + 4x^2 - 7\) is, when you replace \(x\) with 4 or -3 respectively. Although there are several ways to evaluate functions, leveraging techniques like synthetic division can simplify these calculations by streamlining the arithmetic operations involved.

Synthetic Division Method

The synthetic division method is a streamlined version of polynomial long division designed specifically for dividing by a linear binomial, such as \(x - 4\) or \(x + 3\), akin to the scenarios in the given exercise. This method is particularly useful because it limits the use of variables, focusing on coefficients instead and is thus much less cumbersome than traditional long division.

To perform synthetic division:

• List the coefficients of the polynomial, dropping any terms with zero coefficients as necessary.
• Place the value of \(x\) (4 or -3 in the exercise) to the left which is used as the divisor.
• Bring down the leading coefficient to start the process, then multiply it by the divisor \(x\) value and add it to the next coefficient.
• Repeat this step across all coefficients, each time multiplying the previously calculated value by \(x\) and adding it to the next coefficient.
• The remainder represents the function's value at this certain \(x\).

Using synthetic division similarly for both \(f(4)\) and \(f(-3)\) in the exercise, you will end up with a final row entry that directly gives the result of the function evaluation, proving its efficiency in simplifying otherwise complex calculations.