Problem 51 - Algebra and Trigonometry

Question

\(39-51\) . 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

\(P(0.1) = -8.279\).

Step 1: Set up Synthetic Division

First, you'll set up synthetic division to evaluate \( P(0.1) \) without substituting directly. Write the coefficients of \( P(x)=x^{3}+2x^{2}-3x-8 \) as a row: \([1, 2, -3, -8]\). On the left, write the value of \( c \) which is 0.1.

Step 2: Perform Synthetic Division

Bring down the first coefficient (1) to the bottom row. Multiply it by 0.1 and write the result under the second coefficient (2). Add these numbers: 2 + (1 * 0.1) = 2.1 and write it on the bottom row. Repeat for each pair of coefficients: - Multiply 2.1 by 0.1 and add to -3 to get -2.79. - Multiply -2.79 by 0.1 and add to -8 to get -8.279.

Step 3: Interpret the Remainder

The last number in the bottom row of synthetic division is the remainder. This remainder represents \(P(c)\) according to the Remainder Theorem. Thus, \(P(0.1) = -8.279\).

Key Concepts

Polynomial Evaluation

Polynomial evaluation is a method used to find the value of a polynomial at a specific point, in this case, for a given number \( c \). Instead of plugging this number directly into the polynomial equation, you can use more efficient techniques like synthetic division. This is advantageous especially for higher-degree polynomials, as it simplifies calculations and minimizes errors. To evaluate \( P(x) \) for \( x = 0.1 \), synthetic division allows you to bypass direct substitution by methodically breaking down the process.

• This involves rewriting the polynomial using its coefficients.
• These coefficients are then used in a step-by-step process with the given number \( c \) to arrive at the result.

The result of synthetic division directly gives the value of the polynomial at \( x = c \), which in this problem is \( P(0.1) = -8.279 \). This shows a streamlined approach to polynomial evaluation, incorporating mathematical processes such as addition and multiplication repeatedly.

Remainder Theorem

The Remainder Theorem is a powerful tool in algebra that connects polynomial division with evaluation. It states that if you divide a polynomial \( P(x) \) by \( x - c \), the remainder is exactly \( P(c) \). This theorem not only offers a quick route to polynomial evaluation but also validates the results obtained through synthetic division.

• When we use synthetic division to evaluate \( P(x) \) at \( x = 0.1 \), the process concludes with a remainder.
• According to the Remainder Theorem, this remainder is equivalent to the polynomial evaluated at that point: \( P(0.1) \).

This principle simplifies calculations significantly, as it reduces the workload to calculating the remainder rather than going through the entire polynomial substitution process. Thus, the theorem substantiates the result \( P(0.1) = -8.279 \) and confirms its accuracy without direct computation of all polynomial terms.

Polynomial Coefficients

Polynomial coefficients are the numbers preceding each term in a polynomial expression. They play a crucial role when using synthetic division. For the polynomial \( P(x) = x^3 + 2x^2 - 3x - 8 \), the coefficients are \([1, 2, -3, -8]\). These coefficients represent the impact of each term in defining the shape and value of the polynomial.

• The sequence begins with the coefficient of the highest degree term and follows down to the constant term.
• In operations like synthetic division, these coefficients are systematically manipulated to derive values such as the remainder.

Understanding how to utilize these coefficients efficiently allows us to simplify complex polynomial problems. During synthetic division, each coefficient is individually processed alongside the value \( c \), ultimately producing a remainder that represents the evaluated polynomial \( P(x) \) at that specific point. Knowing how to handle polynomial coefficients is essential in mastering synthetic division and related evaluations.