Problem 41 - Algebra and Trigonometry

Question

Use synthetic division and the Remainder Theorem to evaluate \(P(c) .\) \(P(x)=x^{3}+2 x^{2}-7, \quad c=-2\)

Step-by-Step Solution

Answer

\(P(-2) = -7\)

Step 1: Set up synthetic division

Write down the coefficients of the polynomial \(P(x) = x^3 + 2x^2 + 0x - 7\), which are \(1, 2, 0, -7\). We use \(c = -2\) for the division.

Step 2: Perform synthetic division

Bring down the leading coefficient, which is \(1\). Multiply \(1\) by \(-2\) and add to the next coefficient (\(2\)): \(1 \times -2 + 2 = 0\). Repeat for the rest: \(0 \times -2 + 0 = 0\); \(0 \times -2 - 7 = -7\).

Step 3: Interpret the remainder

The result of the synthetic division gives us a remainder of \(-7\). According to the Remainder Theorem, this is \(P(-2)\).

Key Concepts

Polynomial Evaluation

Polynomial evaluation is like checking how a specific value affects the equation. Here, we're determining the result of a polynomial when a certain number replaces the variable. For this exercise, we have a polynomial function \(P(x) = x^3 + 2x^2 - 7\) to evaluate at \(c = -2\).

Using synthetic division makes this process quicker compared to simply substituting and calculating, especially for complex polynomials. By organizing the coefficients \(1, 2, 0, -7\) and employing \(c = -2\), we streamline the calculation through a step-by-step approach.

• Start by jotting down the polynomial's coefficients.
• Use the divisor, here \(-2\), and begin the process of synthetic division.

This method provides us directly with the remainder, which represents the evaluated polynomial value at \(c = -2\) as implied by the remainder.

Remainder Theorem

The Remainder Theorem is a handy tool that connects polynomial division to function evaluation. It states that the remainder obtained from dividing a polynomial by a binomial \((x - c)\) is precisely the value of the polynomial at that same \(c\).

This theorem simplifies the evaluation process vastly. Instead of substituting and simplifying the expression, you simply divide and find the remainder, which directly gives you \(P(c)\).

• Arrange your polynomial for division by noting its coefficients.
• Apply synthetic division to save time and effort.

Thus, if you perform synthetic division on the polynomial with our given \(c = -2\), the remainder you obtain is \(P(-2) = -7\), verifying the theorem.

Algebraic Expressions

Understanding algebraic expressions is foundational in maneuvering through polynomial problems. An algebraic expression is a combination of variables, coefficients, and constants, involving operations like addition or subtraction.

In the exercise given, \(P(x) = x^3 + 2x^2 - 7\) is an instance of an algebraic expression. Each part is called a term, with coefficients attached to variables and constants standing alone.

• The expression is built from different terms: \(x^3\), \(2x^2\), and \(-7\).
• It's crucial to remember when finding the value of \(P(c)\), each term contributes collectively towards the end result after evaluation.

By mastering the arrangement and interaction of these components, interpreting and solving polynomial problems becomes a more intuitive task, especially with such tools as synthetic division.