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The substitution method is a straightforward technique used in algebra to evaluate polynomials by replacing the variable with a given number. It's like solving a puzzle where each piece represents a value you plug into the equation. Start by identifying the value of the variable you need to substitute. For example, in our exercise, the values are 2 and -1. Next, replace the variable in the polynomial with this number and perform the operations step-by-step.

This method helps us see how the polynomial behaves for different values, making it a very practical tool in algebra and everyday problem-solving. The steps are simple and involve basic arithmetic—squaring numbers, multiplying, and adding/subtracting them. So, it's like a recipe: follow the steps, and you'll get the right result.

To evaluate a polynomial means to find its value for a specific variable. Let's break down the process using our given polynomial \[-2x^2 + 5x - 1\].

First, substitute the value into the polynomial.

• For example, if \( x = 2 \), substitute it into the equation to get \( -2(2^2) + 5(2) - 1 \).
• Next, solve each part separately. Square the 2 to get 4, then multiply by -2 to get -8.
• Calculate \( 5 \times 2 \) to get 10.
• And finally combine all terms: -8 + 10 - 1.
• The final result here is 1, so the value of the polynomial when \( x = 2 \) is 1.

Follow similar steps for other values of x, like -1. Make sure to handle the operations carefully, especially signs with negative numbers, to get the correct result.

This practice helps you understand how polynomials work for any given variable's value.

Algebra often requires you to solve problems one step at a time. This step-by-step approach ensures accuracy and understanding. Let's go through our example step-by-step:

1. Identify the polynomial and the value to substitute. Given: \[-2x^2 + 5x - 1\] and \( x = 2 \).
2. Substitute \( x \) with 2: \[-2(2^2) + 5(2) - 1\]
3. Calculate \( 2^2 \) which is 4.
4. Then, \-2 \times 4 = -8 \.
5. Next, calculate \( 5 \times 2 \) = 10.
6. Combine the terms: \-8 + 10 - 1\.
7. Finally, solve the expression: 1.

Treating each step individually helps ensure you do not miss crucial details. This habit is beneficial in solving more complex polynomial equations and other algebra problems.

Practicing this method will help you become more confident in handling algebraic equations, regardless of their complexity.