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Chapter 4: Problem 32

Find each product. $$ (13 r+2 z)(13 r-2 z) $$

Short Answer

Expert verified
169r^2 - 4z^2

Step by step solution

01

Identify the binomials

The given expression egins with two binomials: eq: (13r + 2z)(13r - 2z)These should be recognized as binomials because they each have two terms separated by an addition or subtraction operator.
02

Use the difference of squares formula

When multiplying binomials in the form (eq: (a + b)(a - b), the result is always of the form eq: a^2 - b^2. Identify a and b eq: the binomials eq: a = 13r,b = 2z.
03

Apply the formula

eq: a^2 - b^2, and substitute 13r for a and 2z for beq: (13r)^2 - (2z)^2 which simplifies to: 169r^2 - 4z^2. Therefore the result is 169r^2 - 4z^2.

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Key Concepts

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

Understanding Binomials
In algebra, a binomial is a type of polynomial that contains exactly two terms. These terms are separated by addition or subtraction. For instance, the expressions \(13r + 2z\) and \(13r - 2z\) are binomials because they each have two terms.
  • A term is a single mathematical expression. In \(13r + 2z\), the terms are \(13r\) and \(2z\).
  • The terms can contain variables, coefficients, and constants.
Recognizing binomials is crucial because it allows you to use specific algebraic techniques to manipulate and solve expressions. For example, the difference of squares formula, which we will see in the next section.
Factoring with the Difference of Squares
Factoring is the process of breaking down an expression into simpler components, called factors, that when multiplied together give the original expression. One useful technique is the difference of squares.
  • In this technique, the expression must be in the form \((a + b)(a - b)\).
  • The result is always \(a^2 - b^2\).
Using our example with \(13r\) and \(2z\), you can identify \(a = 13r\) and \(b = 2z\).
Applying the difference of squares formula \[ (13r + 2z)(13r - 2z) = (13r)^2 - (2z)^2 \]
Simplifying each squared term, we get:
  • \((13r)^2 = 169r^2\)
  • \((2z)^2 = 4z^2\)

So the simplified result is \(169r^2 - 4z^2\)
Working with Algebraic Expressions
Algebraic expressions are mathematical phrases that can include numbers, variables, and operation symbols. They represent values and relationships between values. For example, the expression \(13r + 2z\) combines variables \(r\) and \(z\) with coefficients and constants.
  • Variables are symbols used to represent unknown values.
  • Coefficients are numbers multiplied by the variables (e.g., 13 in \(13r\)).
  • Constants are fixed values without variables (e.g., 2 in \(2z\)).

Understanding algebraic expressions is essential because it enables you to manipulate and solve equations. In our exercise, recognizing and simplifying expressions using the difference of squares is an example of this.
By mastering these fundamental concepts, you can tackle more complex problems with confidence.

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