Problem 49 - Precalculus Mathematics for Calculus

Question

Use synthetic division and the Remainder Theorem to evaluate \(P(c)\). $$P(x)=3 x^{3}+4 x^{2}-2 x+1, \quad c=\frac{2}{3}$$

Step-by-Step Solution

Answer

Using synthetic division, \(P\left( \frac{2}{3} \right) = \frac{7}{3}\).

Step 1: Set Up Synthetic Division

To use synthetic division, start by setting up an array. The coefficients from the polynomial \(P(x)=3x^3 + 4x^2 - 2x + 1\) are **3, 4, -2, 1**. Use \(c = \frac{2}{3}\) which will be written outside of the bracket.

Step 2: Write Initial Coefficients

Write down the coefficients in a row: **3, 4, -2, 1**. To the left, write \(c = \frac{2}{3}\).

Step 3: Carry Down the First Coefficient

Carry down the first coefficient (3) in our row to the bottom row to begin our calculations.

Step 4: Multiply and Add to Find Next Coefficient

Multiply \(c = \frac{2}{3}\) by the number carried down (3) to get 2. Add this to the next coefficient (4) in the top row. This gives: \(4 + 2 = 6\).

Step 5: Repeat Multiply and Add Process

Carry down the 6. Multiply 6 by \(\frac{2}{3}\) to get 4. Add this to the next coefficient (-2) : \(-2 + 4 = 2\).

Step 6: Continue the Pattern

Carry down the 2. Multiply 2 by \(\frac{2}{3}\) to get \(\frac{4}{3}\). Add this to the last coefficient (1) to get \(1 + \frac{4}{3} = \frac{7}{3}\).

Step 7: Extract the Remainder

The final value, \(\frac{7}{3}\), represents the remainder. According to the Remainder Theorem, this is the value of \(P\left( \frac{2}{3} \right)\).

Key Concepts

Synthetic Division

Synthetic division is a simplified form of polynomial division, particularly useful when dividing by linear factors. It makes the process less tedious compared to traditional long division with polynomials. Instead of writing all terms fully, you only need to work with the coefficients.
To perform synthetic division:

• Write the coefficients of the polynomial in descending order of power.
• Place the value of the divisor zero (in this case, \( c = \frac{2}{3} \)) outside the bracket.
• Start with bringing down the leading coefficient as the first number of the bottom row.

Through this method, calculations involve straightforward multiplication and addition, reducing the chance of errors and allowing quicker results.

Remainder Theorem

The Remainder Theorem is quite a powerful tool when dealing with polynomial division. It states that the remainder of the division of a polynomial \( P(x) \) by a linear polynomial \( x-c \) is simply \( P(c) \). This provides a fast way to evaluate polynomials at a specific point.
Using synthetic division in this way helps you directly find the value of \( P(c) \) without long calculations. When you are done with the division, the last number in the bottom row is your remainder, which, as per the theorem, is exactly \( P(\frac{2}{3}) \) in this exercise.

Polynomial Evaluation

Evaluating a polynomial at a specific value means calculating the result when a certain number replaces the variable \( x \). A straightforward way is direct substitution, but this can be computationally intensive, especially for higher-degree polynomials.
Synthetic division with the Remainder Theorem offers a speedy alternative for these evaluations, as you only handle the coefficients, making the process more efficient.
In the exercise, we evaluated the polynomial \( P(x) = 3x^3 + 4x^2 - 2x + 1 \) at \( c = \frac{2}{3} \) using these advanced techniques to extract the desired value efficiently.

Coefficients

Coefficients in a polynomial are the numbers in front of the variables, representing a critical part of its structure. For example, in the polynomial given: 3, 4, -2, and 1 are the coefficients.
These coefficients are removed from the variable context and used in synthetic division to perform calculations easily. They help track how the polynomial's terms interact during the division process efficiently.
Without the coefficients, it would be challenging to conduct operations that allow us to find the value of the polynomial at specific points through division methods.