Problem 42 - Algebra and Trigonometry

Question

\(39-51\) . Use synthetic division and the Remainder Theorem to evaluate \(P(c) .\) $$ P(x)=x^{3}-x^{2}+x+5, \quad c=-1 $$

Step-by-Step Solution

Answer

The remainder is 2, so \(P(-1) = 2\).

Step 1: Set Up Synthetic Division

Write down the coefficients of the polynomial \(P(x) = x^3 - x^2 + x + 5\), which are \([1, -1, 1, 5]\). Then, write \(c = -1\) on the left side.

Step 2: Bring Down the Leading Coefficient

Bring down the leading coefficient \(1\) to the bottom row. This will be the starting point for synthetic division.

Step 3: Multiply and Add

Multiply \(-1\) (the value of \(c\)) by the number just brought down (\(1\)) and write the result below the next coefficient. Add this result to the next coefficient \(-1\) to get \(-2\).

Step 4: Repeat the Process

Repeat the multiply and add process for the remaining coefficients. Multiply \(-1\) by \(-2\), write the result (\(2\)) below the next coefficient \(1\), and add to get \(3\). Multiply \(-1\) by \(3\) and add to the last coefficient \(5\) to get \(2\).

Step 5: Interpret the Result

The last number in the bottom row of the synthetic division is \(2\), which is the remainder. According to the Remainder Theorem, \(P(-1) = 2\).

Key Concepts

Polynomial Evaluation

Polynomial evaluation involves determining the value of a polynomial function at a specific point. In this context, the given polynomial is \( P(x) = x^3 - x^2 + x + 5 \). The task is to find the value of \( P(x) \) when \( x = -1 \).

This is often done by substituting \(-1\) in place of \(x\) in the polynomial equation and solving it. However, synthetic division offers a systematic method to achieve this quickly and is particularly useful for evaluating polynomials without needing to substitute and compute each term separately.

Through synthetic division, we can organize and simplify the calculation process. It helps in breaking down the steps, resulting in reduced errors and faster computation.

Remainder Theorem

The Remainder Theorem is a useful principle in algebra, particularly when dealing with synthetic division.

This theorem states that when a polynomial \( P(x) \) is divided by a linear divisor of the form \( x - c \), the remainder of this division is equal to \( P(c) \). For example, in the given exercise, \( P(x) = x^3 - x^2 + x + 5 \) and \( c = -1 \). Using the Remainder Theorem, we processed the polynomial with synthetic division and found the remainder to be \( 2 \).

This remainder tells us that \( P(-1) = 2 \), which means that the polynomial evaluates to 2 at \( x = -1 \).

• Verification of polynomial values becomes much simpler with this theorem.
• It enables predictions about polynomial behavior without full division computations.

Polynomial Coefficients

Polynomial coefficients are the numerical values in front of each term in a polynomial. For the polynomial \( P(x) = x^3 - x^2 + x + 5 \), the coefficients are \([1, -1, 1, 5]\).

When using synthetic division, these coefficients are pivotal as they form the grid where calculations are performed. Here's how they play a role:

• The first coefficient, \(1\), is directly brought down as is.
• Each subsequent coefficient is adjusted through the synthetic division process by multiplying the previous result with \(c = -1\) and then adding them to the next coefficient.

Understanding these coefficients and their manipulation through synthetic division is crucial for accurate polynomial evaluation. Each step with coefficients helps reveal both the quotient (if needed) and the remainder, aiding direct application of the Remainder Theorem.