Problem 42 - Algebra and Trigonometry

Question

Use synthetic division and the Remainder Theorem to evaluate \(P(c) .\) \(P(x)=2 x^{3}-21 x^{2}+9 x-200, \quad c=11\)

Step-by-Step Solution

Answer

The value of \( P(11) \) is 20.

Step 1: Set Up Synthetic Division

To perform synthetic division, we need to use the value of \( c \) which is given as 11. We also need the coefficients of \( P(x) \), which are 2, -21, 9, and -200. Place these coefficients in a row and write 11 to the left side to perform the division.

Step 2: Bring Down the First Coefficient

The first coefficient is 2. Bring it straight down. This will be our starting point for the calculations.

Step 3: Multiply and Add

Multiply the number brought down (2) by \( c \) (which is 11) and add the result to the next coefficient:\[2 \times 11 = 22\]Add this to the second coefficient (-21):\[-21 + 22 = 1\]

Step 4: Repeat the Process

Continue the process. Multiply the newly obtained sum (1) by 11 and add it to the next coefficient:\[1 \times 11 = 11\]Add this to the third coefficient (9):\[9 + 11 = 20\]

Step 5: Final Multiplication and Addition

Repeat again with the new sum found (20). Multiply it by 11 and add to the last coefficient:\[20 \times 11 = 220\]Add this to the last coefficient (-200):\[-200 + 220 = 20\]

Step 6: Interpret the Remainder

The remainder from this division is 20. According to the Remainder Theorem, this remainder is \( P(11) \), meaning it is the value of the polynomial when \( x = 11 \).

Key Concepts

Polynomial Evaluation

Evaluating a polynomial involves finding the value of the polynomial for a specific value of the variable. In this exercise, we need to determine the value of the polynomial \( P(x) = 2x^3 - 21x^2 + 9x - 200 \) when \( x = 11 \). A traditional method to evaluate this is by directly plugging in \( 11 \) for \( x \) and computing the value step by step. However, this can become cumbersome, especially for polynomials of higher degrees.

Instead, a more efficient method known as synthetic division can be used. This approach simplifies the process by using the coefficients of the polynomial and systematically performing operations that mimic long division. It is notably quicker and less prone to error compared to direct substitution, especially with large coefficients or higher-degree terms.

• The first step involves setting up a row of coefficients.
• Next, initiate the division by bringing down the first coefficient.
• Proceed with iterative multiplication and addition.

This streamlined process culminates in identifying the value of the polynomial at a given point, aligned with what is known as the Remainder Theorem.

Remainder Theorem

The Remainder Theorem is an essential tool in algebra that relates polynomial division to polynomial evaluation. In essence, it states that if a polynomial \( P(x) \) is divided by \( x - c \), the remainder of this division is equivalent to \( P(c) \), the value of the polynomial at \( x = c \). This is not only a theoretical concept but a practical strategy for evaluating polynomials without needing to manually substitute values.

For example, when performing synthetic division on our polynomial \( P(x) = 2x^3 - 21x^2 + 9x - 200 \) using \( c = 11 \), the remainder we obtain after completing the division process is exactly \( P(11) \). This method provides both efficiency and accuracy, making it a preferred technique in many algebra problems.

• The theorem links polynomial division operations directly to evaluation.
• It allows for quick calculation of polynomial values, streamlining problem-solving.

Embracing the Remainder Theorem helps in streamlining what would otherwise be labor-intensive processes of substitution and direct computation.

Polynomial Coefficients

Polynomial coefficients are the numerical factors in the terms of a polynomial. In the polynomial \( P(x) = 2x^3 - 21x^2 + 9x - 200 \), the coefficients are 2, -21, 9, and -200. Each coefficient corresponds to a specific term in the polynomial and plays a crucial role in defining the polynomial's behavior.

When applying synthetic division, these coefficients are used directly in the process. The operation involves:

• Listing the coefficients in decreasing order of the power of \( x \).
• Using these coefficients to perform systematic calculations.

The position of each coefficient corresponds to the degree of \( x \) it is attached to. The highest power comes first, moving sequentially to the lowest, which can be a constant or zero power term. Recognizing the placement and role of these coefficients is key in successfully executing synthetic division and, subsequently, in understanding polynomial functions."