Problem 8 - Elementary Algebra

Question

Evaluate. \(5 x 2-2 x+4,\) where \(x=-1\)

Step-by-Step Solution

Answer

The value of the expression is 11.

Step 1: Substitute the value of x

The first step in solving the expression \(5x^2 - 2x + 4\) is to substitute \(x = -1\) into the expression.

Step 2: Calculate the value of x squared

Calculate \((-1)^2 = 1\). This simplifies the expression to \(5(1) - 2(-1) + 4\).

Step 3: Simplify multiplication and negation

Perform the multiplication: \(5 \times 1 = 5\) and \(-2 \times -1 = 2\). The expression now becomes \(5 + 2 + 4\).

Step 4: Add the terms

Add the remaining terms together: \( 5 + 2 + 4 = 11\). Thus, the value of the expression is \(11\).

Key Concepts

Substitution Method

The substitution method is a powerful tool in mathematics, specifically when dealing with algebraic expressions. When you have an expression like \(5x^2 - 2x + 4\), and you're asked to find its value for a specific number, like \(x = -1\), substitution is your friend. Here's how it works:

• Take the value given for \(x\) and replace every instance of \(x\) in the expression with this number.
• Often, you will find parentheses helpful to keep track of where substitutions occur, particularly when negative numbers are involved, as they can affect signs.

Following this method ensures a clear and methodical solution path, reducing errors and making complex expressions more manageable.

Polynomial Evaluation

Polynomial evaluation involves determining the value of a polynomial for a particular variable value. In the original exercise, you began with the expression \(5x^2 - 2x + 4\) and were tasked with finding its value when \(x = -1\). Let's break it down:

• Each term in the polynomial is evaluated separately. Begin by identifying the degree of each term and substitute your value for \(x\).
• Calculate powers first. In this example, for \(x^2\), compute \((-1)^2 = 1\).
• The next step is to multiply coefficients by your calculated power of \(x\) for each term. For example, \(5 imes 1\).

By successfully applying these methods, you can tackle even more complex polynomials with confidence.

Simplifying Expressions

Simplifying expressions is about breaking them down to their most basic form. After substituting and evaluating polynomials, like in our example \(5 \times 1 - 2 \times (-1) + 4\), the next step is to simplify. Consider these steps:

• Perform all multiplications to get rid of coefficients attached to variables.
• Address any negative signs, especially when they appear before numbers or variables being multiplied, as this can change the result.
• Finally, perform any addition or subtraction left in the expression. Combine all your results to reach the simplest form.

Simplifying lets you see the big picture. With practice, recognizing these steps makes solving algebraic expressions more intuitive and satisfying. Enjoy watching the problem transform and "click" as it simplifies on paper!