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Chapter 4: Problem 23

Find each product. $$ (k+5)(k-5) $$

Short Answer

Expert verified
k^2 - 25

Step by step solution

01

Identify the Formula

Recognize that the expression \((k+5)(k-5)\) is a difference of squares. The difference of squares formula is given by \((a+b)(a-b) = a^2 - b^2\).
02

Substitute the Values

Substitute \(a = k\) and \(b = 5\) into the difference of squares formula: \((k+5)(k-5) = k^2 - 5^2\).
03

Simplify the Expression

Simplify the expression, \(k^2 - 5^2\), to get \(k^2 - 25\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Algebraic Expressions
To understand the problem, let's start with algebraic expressions. These are combinations of variables, numbers, and operations. For example, in \(k+5\), \(k\) is a variable, and \(5\) is a constant number. Algebraic expressions can represent real-world problems and help solve them efficiently. You can add, subtract, multiply, and divide them.
In our exercise, \(k+5\) and \(k-5\) are algebraic expressions. When multiplied, they use a special property called the difference of squares. This will simplify the multiplication process.
Once we recognize this property, the rest of the work becomes easier and more systematic.
Factoring
Factoring is breaking down an expression into simpler 'factors' that, when multiplied, give the original expression. For example, consider \(6\). You can write it as \(2 \cdot 3\). This is a simple illustration of factoring.
In our exercise, we're using the difference of squares formula. This formula states \((a+b)(a-b) = a^2 - b^2\). Here, the expression \(k+5\) can be viewed as \(a+b\), and \(k-5\) as \(a-b\). By recognizing this pattern, you can factor the product directly to \k^2 - 25\.
This method of factoring saves time and simplifies the problem efficiently.
Simplification
Simplification is the process of making an expression easier to understand or work with. This often involves combining like terms or using formulas. In our exercise, simplification involves rewriting the product of \(k+5\) and \(k-5\) using the difference of squares formula.
Once we've identified the formula \(a^2 - b^2\), we substitute \(a = k\) and \(b = 5\). This gives us \k^2 - 5^2\. It's important to complete the simplification by calculating \5^2\, which equals \25\. Now, you have \k^2 - 25\.
The final expression, \(k^2 - 25\), is simplified and easier to interpret. Simplifying expressions makes them easier to work with and prepare for further calculations or applications.

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Most popular questions from this chapter

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