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Magazine Chapter 1: Problem 91
Evaluate each expression. See Example 10. $$ a^{2}+2 a b+b^{2} \text { for } a=-5 \text { and } b=-1 $$
Short Answer
Expert verified
The expression evaluates to 36.
Step by step solution
01
Identify the Expression
The given expression is \( a^2 + 2ab + b^2 \). We will evaluate this expression for \( a = -5 \) and \( b = -1 \).
02
Substitute Values into the Expression
Substitute \( a = -5 \) and \( b = -1 \) into the expression: \( (-5)^2 + 2(-5)(-1) + (-1)^2 \).
03
Calculate \( a^2 \)
Calculate \((-5)^2\) which equals 25.
04
Calculate \( 2ab \)
Calculate \(2(-5)(-1)\) which equals 10. This is because the product of two negatives is a positive.
05
Calculate \( b^2 \)
Calculate \((-1)^2\) which equals 1.
06
Add the Results
Add the results obtained from Steps 3, 4, and 5: \(25 + 10 + 1 = 36\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
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:
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.
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鈥檚 how you break it down:
Here鈥檚 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.
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:
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: \((-5)^2 = 25\).
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:
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.
