Problem 34 - Precalculus: Functions and Graphs

Question

Use synthetic division to find \(f(c)\). $$f(x)=x^{3}-3 x^{2}-8 ; \quad c=1+\sqrt{2}$$

Step-by-Step Solution

Answer

Use synthetic division with coefficients [1, -3, 0, -8] and divisor 1+√2. Result is f(1+√2).

Step 1: Set Up Synthetic Division

First, we need to write the coefficients of the polynomial function, which are 1 (for \(x^3\)), -3 (for \(x^2\)), 0 (for \(x\) as it is missing), and -8 (constant term). We'll use these elements in synthetic division with the value \(c = 1 + \sqrt{2}\).

Step 2: Start Synthetic Division

Write \(1 + \sqrt{2}\) to the left side of the bracket. Write the coefficients \(1, -3, 0, -8\) inside the bracket for division. Bring down the first coefficient \(1\) directly below the line.

Step 3: Perform the Division

Multiply \(1\) by \(1 + \sqrt{2}\) and write the result under the second coefficient (-3). Calculate and write the new sum: \[-3 + (1 + \sqrt{2}) = -2 + \sqrt{2}\].Repeat the process by multiplying \(-2 + \sqrt{2}\) by \(1 + \sqrt{2}\). Continue this process for all coefficients.

Step 4: Complete the Process

Continue multiplying and adding:- Multiply \(-2 + \sqrt{2}\) by \(1 + \sqrt{2}\), write under \(0\).- Sum: \(0 + (-2 + \sqrt{2})(1 + \sqrt{2})\).- Calculate these expressions to complete the synthetic division.

Step 5: Evaluate and Conclude

The result of the final sum in synthetic division row (below last coefficient) is \(f(c)\). Compute the final value, which represents the polynomial evaluated at \(c = 1 + \sqrt{2}\).

Key Concepts

Polynomial Evaluation

Polynomial evaluation involves finding the value of a polynomial function for a given value of the variable, often denoted as "x". In simpler terms, it is about plugging a specific value into the polynomial equation and calculating the result to see what the polynomial equals at that point.
To evaluate a polynomial like \(f(x) = x^3 - 3x^2 - 8\) at \(c = 1 + \sqrt{2}\), one might use several methods, including direct substitution or synthetic division. Synthetic division is particularly useful because it simplifies the calculation process, especially when dealing with complex or large coefficients.

• In synthetic division, you start with the coefficients of the polynomial.
• Substitute the value of \(c\) into the synthetic division format to compute the value of the polynomial.
• The result from the division provides the value of the polynomial at \(c\), essentially giving us \(f(c)\).

This process is not only efficient but can also be used to verify the correctness of solutions by checking if \(f(c) = 0\), which indicates \(c\) is a root of the polynomial.

Roots of Polynomial

Roots of a polynomial are values for which the polynomial equals zero. They are also known as "zeros" or "solutions" of the polynomial equation. Understanding roots is crucial because they tell us where the polynomial graph intersects the x-axis.
To find the roots of a polynomial, different methods can be used, such as factoring, the quadratic formula, or synthetic division.

• Synthetic division can help in verifying if a guessed root \(c\) is indeed a root by showing if the polynomial evaluates to zero at \(c\).
• If the remainder from synthetic division is zero when evaluating \(f(c)\), \(c\) is definitely a root.
• Discovering roots can aid in further factoring the polynomial to find other roots.

Finding the roots is not only about solving equations but also about understanding the behavior of polynomials and their graphs.

Coefficients in Polynomials

Coefficients in polynomials are the numbers that multiply the variables in the polynomial expression. These coefficients determine the shape and position of the polynomial graph.
When setting up synthetic division, recognizing the coefficients is crucial. In the polynomial \(f(x) = x^3 - 3x^2 - 8\), the coefficients are:

• 1 for \(x^3\)
• -3 for \(x^2\)
• 0 for \(x\) (since there is no \(x\) term)
• -8 as the constant term

The coefficients provide the essential information needed in the synthetic division setup:

• They are arranged in order of decreasing powers of \(x\).
• If a term is missing, it is represented by a zero to maintain the correct order.
• In synthetic division, these coefficients interact with the day-to-day value chosen to evaluate or divide the polynomial, affecting the remainder and quotient calculations.

Understanding coefficients is central to performing polynomial operations like evaluation, division, and finding roots.