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Chapter 7: Problem 274
The height a twelve foot ladder can reach up the side of a building if theladder's base is \(b\) feet from the building is the square root of the binomial\(144-b^{2}\). Factor the binomial.
Short Answer
Expert verified
(144 - b^2) = (12 + b)(12 - b).
Step by step solution
01
- Identify the Binomial
The given binomial is (144 - b^2).
02
- Recognize Difference of Squares Format
The binomial can be recognized as a difference of squares: (a^2 - b^2). Here, a = 12 (since 144 = 12^2) and b = b.
03
- Apply Difference of Squares Formula
Using the difference of squares formula, (a^2 - b^2) = (a + b)(a - b), factor (144 - b^2) = (12 + b)(12 - b).
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Difference of Squares
In algebra, the 'difference of squares' is a special type of binomial. Understanding this concept is essential to factor certain expressions. The formula for this is: \[ a^2 - b^2 = (a + b)(a - b) \]. The given binomial expression in the exercise is (144 - b^2). Here, 144 is a perfect square (since 144 = 12^2). By identifying the values of ‘a’ and ‘b’, we can rewrite our original binomial as a difference of squares. So, letting \(a = 12\) and \(b = b\), it matches the pattern and can be factored using the formula above.
This results in the expression (12 + b)(12 - b).
Remember, recognizing such patterns is a key skill in algebra.
Binomial Expressions
A 'binomial' is an algebraic expression with exactly two terms. Examples include \(x + y\), \(3a - 4b\), and in our case, \(144 - b^2\).
The subtraction in \(144 - b^2\) helps us see it's a specific type known as a 'difference'. Binomial expressions are fundamental in algebra because many complex problems can be simplified into these two-term expressions. To identify a binomial, look for two separate terms joined by an addition or subtraction sign.
The order and operation (addition or subtraction) between these terms are crucial, as they determine how we can factor or otherwise manipulate the expression.
Factoring Techniques
Factoring is the process of breaking down an expression into simpler ‘factors’ or components. One common technique is recognizing a pattern like the difference of squares.
For example, to factor \(144 - b^2\), you recognize the format \(a^2 - b^2\).
This technique involves these steps:
- Identify the pattern (difference of squares in this case).
- Rewrite the numbers as squares (\(144 = 12^2\)).
- Apply the difference of squares formula: \[a^2 - b^2 = (a + b)(a - b)\].
Other factoring techniques include factoring out the greatest common factor (GCF), grouping, and using quadratic equations.
Choosing the right technique depends on the specific details of the algebraic expression.
Algebra Problem Solving
Effective problem-solving in algebra often involves recognizing patterns and knowing which techniques to apply and when.
Start by carefully reading the problem. Identify key components like variables and constants.
In this exercise, the problem provided a binomial and asked us to factor it.
Next, determine the appropriate method. For \(144 - b^2\), recognizing it as a difference of squares was crucial.
Applying the formula lead to the solution: \((12 + b)(12 - b)\).
Algebra problems can range from simple to complex, and building a solid foundation in these techniques makes tackling each problem more manageable.
Practice regularly, and with time, you'll improve your skills in recognizing patterns and applying the right methods.
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