Problem 9 - Algebra and Trigonometry with Analytic Geometry

Question

$$ \text { Use the remainder theorem to find } f(c) \text {. } $$ $$ f(x)=3 x^{3}-x^{2}+5 x-4 ; \quad c=2 $$

Step-by-Step Solution

Answer

The value of \( f(2) \) is 26.

Step 1: Substitute c into f(x)

We are given the function \( f(x) = 3x^3 - x^2 + 5x - 4 \) and the value \( c = 2 \). Using the remainder theorem, we can substitute \( c \) into \( f(x) \) to find \( f(c) \). Let's replace \( x \) with \( 2 \) in the equation.\[f(2) = 3(2)^3 - (2)^2 + 5(2) - 4\]

Step 2: Calculate each term

Now, calculate each term separately:1. \( 3(2)^3 = 3 imes 8 = 24 \)2. \( -(2)^2 = -4 \)3. \( 5(2) = 10 \)4. \( -4 \) remains \( -4 \)

Step 3: Sum the calculated terms

Add up the terms calculated in Step 2:\[f(2) = 24 - 4 + 10 - 4\]First, perform the operation from left to right:1. \( 24 - 4 = 20 \)2. \( 20 + 10 = 30 \)3. \( 30 - 4 = 26 \)

Step 4: Final Result

Thus, by using the remainder theorem and substituting \( c \) into \( f(x) \), we find that \( f(2) = 26 \).

Key Concepts

Polynomial Evaluation

A polynomial is a mathematical expression consisting of variables and coefficients, stitched together using operations like addition, subtraction, and multiplication. Evaluating a polynomial involves finding its value for a particular substitution of the variable. In this context, the polynomial function given is \( f(x) = 3x^3 - x^2 + 5x - 4 \). Evaluating this polynomial at a particular value, such as \( x = 2 \), involves substituting 2 in place of \( x \) and simplifying the expression to determine the result.In polynomial evaluation, you systematically replace all instances of the variable \( x \) with the number you're evaluating at, ensuring you adhere strictly to the order of operations. Remember to:

• First, handle the powers of \( x \) by replacing with the number.
• Second, perform any multiplication.
• Finally, execute all additions and subtractions to reach the simplification.

This ensures that every term of the polynomial is carefully assessed, leading to a correct and precise evaluation of the polynomial.

Substitution in Algebra

Substitution is a fundamental process in algebra that allows you to solve equations and evaluate expressions by replacing variables with their corresponding values. In the context of evaluating the polynomial \( f(x) = 3x^3 - x^2 + 5x - 4 \) at \( x = 2 \), substitution directly helps you plug \( x = 2 \) into the polynomial.This process makes transforming algebraic expressions into numerical values simple and easy. Here’s how you do it:

• Take each instance where the variable \( x \) appears in the polynomial.
• Substitute \( x \) with the specified number—in our example, \( 2 \).
• Calculate each portion of the polynomial to simplify it step by step.

Utilizing substitution simplifies algebra into a series of calculations, providing a clear path to finding the function’s value at specific points. It’s an essential skill that helps in not just evaluating polynomials, but solving many types of algebraic problems.

Finding Function Value

Finding the value of a function at a particular point is a practical application of substitution in a polynomial. For instance, to find \( f(2) \) for the function \( f(x) = 3x^3 - x^2 + 5x - 4 \), you substitute \( x = 2 \) into the polynomial and simplify.Here's the step-by-step approach:

• Replace \( x \) with 2 in each term of the polynomial.
• Compute the numerical value of each modified term.
  For example: \(3(2)^3 = 24\), \(-(2)^2 = -4\), \(5(2) = 10\), and \(-4\).
• Add or subtract each calculated term in sequence to find the result.

In our example, the sum \( 24 - 4 + 10 - 4 \) leads to \( 26 \). So, the value of the function \( f \) at \( x = 2 \) is \( 26 \). This highlights how evaluating a function can converge into a straightforward calculation through the power of substitution and arithmetic.