Problem 91 - Elementary and Intermediate Algebra

Question

Evaluate each expression. See Example 10. $$ a^{2}+2 a b+b^{2} \text { for } a=-5 \text { and } b=-1 $$

Step-by-Step Solution

Answer

The expression evaluates to 36.

Step 1: Identify the Expression

The given expression is \( a^2 + 2ab + b^2 \). We will evaluate this expression for \( a = -5 \) and \( b = -1 \).

Step 2: Substitute Values into the Expression

Substitute \( a = -5 \) and \( b = -1 \) into the expression: \( (-5)^2 + 2(-5)(-1) + (-1)^2 \).

Step 3: Calculate \( a^2 \)

Calculate \((-5)^2\) which equals 25.

Step 4: Calculate \( 2ab \)

Calculate \(2(-5)(-1)\) which equals 10. This is because the product of two negatives is a positive.

Step 5: Calculate \( b^2 \)

Calculate \((-1)^2\) which equals 1.

Step 6: Add the Results

Add the results obtained from Steps 3, 4, and 5: \(25 + 10 + 1 = 36\).

Key Concepts

Substitution Method

The substitution method in algebra is a technique where you replace variables with specific values to simplify and solve expressions or equations. For example, when given the expression \( a^2 + 2ab + b^2 \), you can substitute \( a = -5 \) and \( b = -1 \). Substituting means you directly replace every instance of \( a \) and \( b \) in the expression with \(-5\) and \(-1\), respectively.
This method is incredibly useful in simplifying algebraic expressions, allowing you to convert them into numerical calculations:

• Identify the variables in the expression.
• Replace each variable with its assigned number.
• Perform arithmetic operations to solve.

This technique not only helps in solving problems, but it also enhances understanding by making abstract expressions more tangible through numerical representation.

Polynomial Evaluation

Polynomial evaluation is the process of calculating the value of a polynomial expression for given values of its variables. In the context of our example, the polynomial \( a^2 + 2ab + b^2 \) is evaluated by substituting and simplifying the expression using the given values of \( a \) and \( b \).
Here’s how you break it down:

• Insert the given values into the polynomial.
• Evaluate any operations, especially order of operations like exponents and multiplication, first.
• Add or subtract the resulting values as necessary.

By simplifying, you turn a polynomial into straightforward arithmetic. This process gives you a numerical value, which is the evaluated result of the polynomial for those specific variable values.

Exponents

Exponents are a way to represent repeated multiplication of the same number by itself. In the expression \( a^2 + 2ab + b^2 \), both \( a^2 \) and \( b^2 \) are terms with exponents.
Here's a quick review of how exponents work:

• An exponent indicates how many times a number is multiplied by itself.
• \( a^2 \) means \( a \times a \).
• Negative bases, like \((-5)^2\), require special attention, as they involve considering the negative sign in the multiplication.
• Remember, a negative number squared always results in a positive number: \((-5)^2 = 25\).

Understanding how exponents are calculated is crucial in evaluating polynomial expressions accurately.

Step-by-step Solution

Understanding a solution step-by-step helps demystify the process of solving algebraic expressions. Each step builds on the previous to arrive at the solution logically and systematically.
Here's how you follow a step-by-step solution:

• **Identify the Expression:** Recognize the polynomial and the need for substitution.
• **Substitution:** Insert known values into the expression.
• **Simple Calculations:** Calculate individual components like exponents and products.
• **Final Addition:** Add all parts together for the final result.

Following these steps ensures you handle every part of the expression correctly, ultimately leading you to the right answer, which in this case is 36. This approach is critical not only for understanding current problems but for building skills to tackle new, similar challenges.