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Chapter 0: Problem 41

Perform the indicated operations and simplify. $$ (4 x-5)(4 x+5) $$

Short Answer

Expert verified
16x^2 - 25

Step by step solution

01

- Recognize the pattern

This expression is a difference of squares: (a - b)(a + b) = a^2 - b^2.
02

- Identify the values of a and b

In this case, let a = 4x and b = 5.
03

- Apply the difference of squares formula

Using the values from Step 2, apply the formula a^2 - b^2: \[(4x)^2 - 5^2\].
04

- Simplify the expression

Calculate the squares and simplify: a^2 - b^2 = \[16x^2 - 25\].

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Key Concepts

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

Algebraic Expressions
Algebraic expressions involve variables, constants, and operations. These are the building blocks of algebra. For instance, in the expression \(4x - 5) \) and \(4x + 5)\), '4x' and '5' are terms.
Variables vary and are usually represented by letters like 'x' or 'y'. Constants stay the same and are typically numbers. Operations can include addition, subtraction, multiplication, and division.
When working with expressions, combining like terms and following the order of operations is crucial.
Recognizing patterns like the difference of squares can simplify the process significantly.
Binomial Multiplication
Binomial multiplication involves multiplying two expressions of the form \( (a \pm b)\). A common way to do this is the FOIL method, which stands for First, Outer, Inner, Last.
For example, consider \( (4x - 5)(4x + 5 ) \). Each term in the first binomial multiplies by each term in the second binomial.
  • First: \(4x \cdot 4x \)
  • Outer: \(4x \cdot 5 \)
  • Inner: \(-5 \cdot 4x\)
  • Last: \(-5 \cdot 5\)

In the case of the difference of squares, \((a - b)(a + b) = a^2 - b^2\). It's a shorthand that saves you time!
Simplifying Expressions
Simplifying means making an expression easier to work with. It involves combining like terms and following mathematical rules.
In the example given, \((4x - 5)(4x + 5)\), we use the difference of squares formula to get \(16x^2 - 25\).
First, recognize that \( (4x - 5)(4x + 5) \) follows the pattern \((a - b)(a + b) = a^2 - b^2\). Then identify 'a' and 'b' where \( a = 4x \ and \ b = 5 \).
Next, apply the formula: \(a^2 - b^2 \rightarrow (4x)^2 - 5^2 \rightarrow 16x^2 - 25 \). This simplified form is much easier to work with in further calculations.

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