Problem 35 - Precalculus with Calculus Previews

Question

Use synthetic division and the Remainder Theorem to find \(f(c)\) for the given value of c. $$ f(x)=2 x^{3}-3 x^{2}-8 x+6 ; c=\sqrt{2} $$

Step-by-Step Solution

Answer

\(f(\sqrt{2}) = -2\sqrt{2} + 4\)

Step 1: Set Up the Synthetic Division

First, identify the value of \(c = \sqrt{2}\) which will be used in the synthetic division process. The coefficients of the polynomial \(f(x) = 2x^3 - 3x^2 - 8x + 6\) are 2, -3, -8, and 6. Write these coefficients in a row, to the right of which you will perform the synthetic division.

Step 2: Perform the Synthetic Division

Start by bringing down the first coefficient (2) as it is. Multiply this by \(c = \sqrt{2}\) and write the result underneath the next coefficient (-3). Add the two numbers: -3 + (2\sqrt{2}). Repeat these steps, always multiplying the result of the addition by \(\sqrt{2}\), and add to the next coefficient until you reach the end.

Step 3: Calculate \(f(\sqrt{2})\)

The final value after all additions in the synthetic division process is the remainder, which according to the Remainder Theorem, equals \(f(\sqrt{2})\). Perform the final calculations to find this value. The intermediate calculations involve handling expressions like \((2 + \sqrt{2})(\sqrt{2})\) which simplify to \(2\sqrt{2} + 2\), and continuing this process yields the remainder as \(f(\sqrt{2}) = -2\sqrt{2} + 4\).

Key Concepts

Remainder Theorem

The Remainder Theorem is a very useful tool in algebra, specifically when working with polynomials. It provides a quick way to evaluate a polynomial at a given point without directly substituting the value into the polynomial itself.
Simply put, if you divide a polynomial \( f(x) \) by a linear divisor \( x - c \), the remainder of this division is equal to \( f(c) \).
This is handy because performing polynomial division is generally less cumbersome than plugging in complex values directly into the polynomial function.
For instance, rather than substituting \( \sqrt{2} \) into \( f(x) = 2x^3 - 3x^2 - 8x + 6 \), you can use synthetic division to streamline the process and directly find the remainder, which is \( f(\sqrt{2}) \).

• The Remainder Theorem simplifies calculations, especially with irrational or complex numbers.
• It transforms evaluation into a division problem, reducing potential arithmetic errors.

By using the Remainder Theorem, evaluating polynomials becomes much more approachable and efficient, especially under time constraints such as exams.

Polynomial Division

Polynomial division, much like regular numeric division, helps us find how many times a polynomial, known as the divisor, fits into another polynomial, known as the dividend.
Now, there are primarily two methods for dividing polynomials: long division and synthetic division. Synthetic division is particularly efficient for dividing polynomials by a linear factor.
It works best when the divisor is in the form \( x - c \), where \( c \) is a constant.
Instead of manipulating variable terms like in long division, synthetic division simplifies the process using coefficients only. Here’s a brief look at how it is carried out:

• List the coefficients of the polynomial.
• Use the constant \( c \) (from \( x - c \)).
• Carry out a sequence of multiplications and additions to find the remainder and quotient.

For instance, with \( f(x) = 2x^3 - 3x^2 - 8x + 6 \) and \( c = \sqrt{2} \), synthetic division provides a systematic way to find \( f(\sqrt{2}) \).
This method also aids students in avoiding the pitfalls of minor calculation mistakes by providing a clear structure to follow.

Algebraic Functions

Algebraic functions are equations that include polynomials, roots, or a combination of both. They represent a wide variety of equations you'll encounter in mathematics.
These functions form the backbone of many algebra problems and are typically expressed in the format \( f(x) = \) some expression involving \( x \).
In algebra, we often manipulate these functions to analyze, simplify, and solve equations. Here’s what's important to keep in mind:

• Understanding how to work with polynomials, including operations like addition, subtraction, and especially division, enhances your grip on algebraic functions.
• Using techniques like synthetic division can simplify solving equations with roots as variables, a common scenario in advanced algebra.

For our polynomial \( f(x) = 2x^3 - 3x^2 - 8x + 6 \), and the task to find \( f(\sqrt{2}) \), understanding both polynomials and their operations, including division, enables you to efficiently solve such equations by using synthetic division and the Remainder Theorem described earlier.
This shows how powerful algebraic functions are when combined with a good strategy and the right tools.