Problem 49 - Algebra and Trigonometry

Question

Use synthetic division and the Remainder Theorem to evaluate \(P(c) .\) \(P(x)=x^{3}+2 x^{2}-3 x-8, \quad c=0.1\)

Step-by-Step Solution

Answer

The value of \(P(0.1)\) is \(-8.279\).

Step 1: Understand the problem statement

We are given the polynomial \(P(x) = x^3 + 2x^2 - 3x - 8\) and a value \(c = 0.1\). Our goal is to evaluate \(P(c)\) using synthetic division and the Remainder Theorem.

Step 2: Arrange the polynomial for synthetic division

Write the coefficients of the polynomial \(P(x) = x^3 + 2x^2 - 3x - 8\) in a row: \(1, 2, -3, -8\). We will use \(c = 0.1\) as the divisor.

Step 3: Perform synthetic division

1. Bring down the first coefficient: \(1\).2. Multiply it by \(0.1\) and add to the next coefficient: - \(1 \times 0.1 = 0.1\). - \(2 + 0.1 = 2.1\).3. Multiply \(2.1\) by \(0.1\) and add to the next coefficient: - \(2.1 \times 0.1 = 0.21\). - \(-3 + 0.21 = -2.79\).4. Multiply \(-2.79\) by \(0.1\) and add to the last coefficient: - \(-2.79 \times 0.1 = -0.279\). - \(-8 - 0.279 = -8.279\).

Step 4: Recognize the remainder

The remainder from the synthetic division process is \(-8.279\). According to the Remainder Theorem, \(P(c) = -8.279\) when \(c = 0.1\).

Key Concepts

Polynomial Evaluation

The core of evaluating a polynomial is substituting a specific value for the variable and simplifying the expression. Typically, you would plug the number directly into the polynomial and work through the arithmetic. However, techniques like synthetic division can make this process much more efficient, especially for polynomials of higher degrees. Given the polynomial \(P(x) = x^3 + 2x^2 - 3x - 8\) and the value \(c = 0.1\), our objective is to find \(P(0.1)\). Instead of manually inserting \(0.1\) into every \(x\) and calculating, synthetic division provides a structured approach, leading us directly to the value of \(P(0.1)\). By organizing and methodically working with the coefficients of the polynomial, you streamline the evaluation process, gaining not only the result but also deeper insights into the polynomial's behavior.

Remainder Theorem

The Remainder Theorem is a powerful tool in algebra that connects polynomial division to polynomial evaluation. It states that if you divide a polynomial \(P(x)\) by \(x-c\), then the remainder of this division is \(P(c)\). In the context of the given problem, where \(P(x) = x^3 + 2x^2 - 3x - 8\) and \(c = 0.1\), applying synthetic division leads us to a remainder of \(-8.279\). Thus, the Remainder Theorem tells us directly that \(P(0.1) = -8.279\). This theorem not only provides a shortcut in polynomial evaluation but also serves as a vital concept for understanding the relationship between roots, factors, and values of polynomials.

Coefficients in Synthetic Division

Synthetic division utilizes the coefficients of a polynomial to simplify division, making it quicker and less error-prone than traditional methods. For \(P(x) = x^3 + 2x^2 - 3x - 8\), the coefficients \(1, 2, -3, -8\) are used. You systematically bring these coefficients into the division process while multiplying by \(c = 0.1\) at each step.

• Start with the first coefficient.
• Multiply by \(c\), then add it to the next coefficient.
• Continue this pattern to process all coefficients.

This strategic organization of calculations reduces complexity and enhances accuracy, showcasing the strength of synthetic division both in theory and practice. Remember, it's essential to align the coefficients precisely, ensuring each aligns with its corresponding power of \(x\), for the division to yield correct results.