Problem 90 - Intermediate Algebra

Question

Let \(P(x)=x^{3}-6 x^{2}-9 x+4 .\) You now know two ways to find \(P(6) .\) What are they? Which method do you prefer?

Step-by-Step Solution

Answer

Two ways are synthetic substitution and direct substitution. I prefer synthetic substitution for its efficiency.

Step 1: Evaluate Directly with Synthetic Substitution

Synthetic substitution is a quick way to evaluate a polynomial function at a given point. To find \( P(6) \), set up the synthetic division using 6. Use the coefficients of \( P(x) = x^3 - 6x^2 - 9x + 4 \) which are 1, -6, -9, and 4.- Drop down the first coefficient: **1**.- Multiply 6 by 1 (the current value) and add it to the next coefficient: **6 \( * 1 \) + (-6) = 0**.- Multiply 6 by 0 (the current value) and add it to the next coefficient: **6 \( * 0 \) + (-9) = -9**.- Multiply 6 by -9 (the current value) and add it to the last coefficient: **6 \( * (-9) \) + 4 = -50**.Thus, \( P(6) = -50 \).

Step 2: Evaluate Using the Polynomial Function Directly

Substitute \( x = 6 \) directly into the polynomial \( P(x) = x^3 - 6x^2 - 9x + 4 \).- Calculate \( 6^3 = 216 \).- Calculate \(-6 \times 6^2 = -6 \times 36 = -216 \).- Calculate \(-9 \times 6 = -54 \).- Add these values together with the constant term: \( 216 - 216 - 54 + 4 = -50 \).Thus, \( P(6) = -50 \).

Step 3: Compare the Methods and Choose a Preference

Method 1, synthetic substitution, is often faster and less prone to calculation error for larger polynomials compared to substituting directly and calculating each term. However, Method 2 might be favored if looking for insight into polynomial structure as each term is explicitly computed. Personally, I prefer Method 1, synthetic substitution, due to its efficiency and simplicity in handling arithmetic operations.

Key Concepts

Synthetic Substitution

Synthetic substitution is a streamlined method used to evaluate polynomial functions at a specific value quickly. It's especially handy for polynomials of higher degrees. Instead of substituting a value directly into the polynomial, you use a process similar to synthetic division.
- **Setup:** Begin by writing down the coefficients of the polynomial in order. For the polynomial \( P(x) = x^3 - 6x^2 - 9x + 4 \), those coefficients are \( 1, -6, -9, \) and \( 4 \).
- **Process:** The next step involves using the value you want to substitute (in this case, \( x = 6 \)) to perform a series of arithmetic operations. Start by dropping the first coefficient straight down as the initial result.
- **Multiplication and Addition:** Multiply the result by the substitution value, and add to the next coefficient. This process continues until all coefficients have been used.
- **Result:** The final number you reach is the value of the polynomial evaluated at \( x \). For our example, we found that \( P(6) = -50 \).
This method is favored for its ability to minimize errors and reduce computation time, as you deal directly with the coefficients and avoid recalculating each polynomial power and multiplication separately.

Polynomial Function

A polynomial function is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. The concept can be illustrated with the example polynomial \( P(x) = x^3 - 6x^2 - 9x + 4 \).
- **Degree:** The degree of a polynomial is determined by the highest power of its variable, which in this case is 3 (as seen in \( x^3 \)).
- **Coefficients:** These are the numbers in front of each power of \( x \). In \( P(x) \), the coefficients are \( 1, -6, -9, \) and \( 4 \). Each represents a linear multiplier for that term in the polynomial.
- **Evaluation:** The process of determining the value of a polynomial for a specific value of \( x \) (a process we call evaluating the polynomial) is done by substituting the value into the polynomial expression.
Polynomials are essential in various fields of mathematics and science because they can model different kinds of behavior, from simple linear relationships to more complex quadratic and cubic trends.

Arithmetic Operations

Arithmetic operations form the core of any polynomial evaluation method, whether you're using synthetic substitution or direct substitution.
- **Addition and Subtraction:** These operations combine different terms of the polynomial when evaluating or simplifying expressions.
- **Multiplication:** Involves multiplying coefficients and variables raised to a power by the value of \( x \) being evaluated, as seen in \( 6 \, \times \, x \) parts of these processes.
- **Order of Operations:** Ensure correctness by following proper arithmetic rules, such as handling coefficients before dealing with exponents, and keeping track of positive and negative signs. These execute precisely as given before summing or subtracting the result.
- **Streamlining Calculation:** In synthetic substitution, ordered operations allow for faster computation with fewer steps, showing its efficiency over explicitly dealing with each power of \( x \) separately.
Mastering these operations is critical to accurately evaluate polynomials and solve algebraic expressions efficiently, reducing the risk of errors during calculations.