Problem 29 - Precalculus: Functions and Graphs

Question

Use synthetic division to find \(f(c)\). $$f(x)=2 x^{3}+3 x^{2}-4 x+4 ; \quad c=3$$

Step-by-Step Solution

Answer

The value of \(f(3)\) is 73.

Step 1: Set Up Synthetic Division

To use synthetic division, set up the "division box". Write down the dividend's coefficients: \(2, 3, -4, 4\). Next, place the value of \(c\), which is \(3\), to the left of the division box.

Step 2: Bring Down the Leading Coefficient

Bring the leading coefficient of the dividend, which is \(2\), straight down below the line. This will be the first value in your solution row.

Step 3: Multiply and Add

Multiply \(3\) (the value of \(c\)) by the value on the solution row directly above (\(2\) initially), which equals \(6\). Place this \(6\) below the second coefficient in the dividend. Add the second coefficient \(3\) (of \(f(x)\)) to this \(6\), yielding \(9\).

Step 4: Repeat Multiplication and Addition

Multiply \(3\) by \(9\), which equals \(27\), and place it below the next coefficient (\(-4\)). Add the result to \(-4\) to get \(23\).

Step 5: Final Multiplication and Addition

Multiply \(3\) by the most recent result, \(23\), resulting in \(69\). Add this \(69\) to the last coefficient \(4\), yielding \(73\). This result, \(73\), is the remainder, which also equals \(f(3)\).

Key Concepts

Polynomial Functions

Polynomial functions are expressions involving variables and coefficients, combined using addition, subtraction, multiplication, and non-negative integer exponents. They can be as simple as a constant or as complex as a high-degree expression. The degree of a polynomial, which is the highest power of the variable present, plays a critical role in determining the behavior and roots of the function.
Common characteristics of polynomial functions include:

• A polynomial with one term is called a monomial.
• A polynomial with two terms is a binomial.
• A polynomial with three terms is a trinomial.

For instance, in our example, the polynomial function is given by:
\[ f(x) = 2x^3 + 3x^2 - 4x + 4 \] Here, the highest degree is 3, and it contains four terms with coefficients 2, 3, -4, and 4 respectively.
Polynomials have continuous graphs and smooth curvatures, making them well-behaved mathematical objects to work with in calculus and algebra.

Remainder Theorem

The remainder theorem provides a handy way to find the remainder of a polynomial division without performing long division. The theorem states that if a polynomial \(f(x)\) is divided by \(x - c\), the remainder of this division is \(f(c)\).
In simple terms:

• Substitute \(c\) into the polynomial \(f(x)\).
• The result is the remainder of the division.

Using synthetic division, as shown in the example, we can efficiently find \(f(c)\), where \(c\) is the value being substituted. In our problem, dividing \(f(x) = 2x^3 + 3x^2 - 4x + 4\) by \(x - 3\) gives us the remainder of 73, meaning \(f(3) = 73\).
This theorem not only saves time but also plays a key role in topics like factor theorem and testing potential roots of polynomial equations.

Evaluation of Functions

Evaluating functions means finding the output value of a function for a given input. In the context of polynomial functions, it involves substituting numerical values into the expression.
For example, to find \(f(3)\) for the polynomial function \(f(x) = 2x^3 + 3x^2 - 4x + 4\), we substitute \(x = 3\) into each term:

• Calculate each term individually: \((2 \times 3^3), (3 \times 3^2), (-4 \times 3), + 4\).
• Add them all together.

This straightforward process ensures finding exact function values at specific points.
Additionally, synthetic division, as demonstrated, streamlines this evaluation, particularly when calculating points for polynomial functions. Here, the efficient calculation yields \(f(3) = 73\), with synthetic division's compact arrangement helping avoid manual errors, especially with larger polynomials.