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Chapter 5: Problem 14

Multiply. $$(4 x-5)(4 x+5)$$

Short Answer

Expert verified
The solution is \( 16x^2 - 25 \).

Step by step solution

01

Identify the Binomial Formulas

Notice that the expressions \((4x-5)\) and \((4x+5)\) are in the form of \((a-b)(a+b)\). This suggests that the difference of squares formula can be used.
02

Apply the Difference of Squares Formula

The difference of squares formula is \(a^2 - b^2\). Here, \(a = 4x\) and \(b = 5\). Substitute these into the formula to get \( (4x)^2 - 5^2 \).
03

Simplify the Expression

Calculate \( (4x)^2 \) and \( 5^2 \). \( (4x)^2 = 16x^2 \) and \( 5^2 = 25 \). Therefore, the expression becomes \ 16x^2 - 25 \.
04

Write the Final Answer

After simplification, the final answer is \( 16x^2 - 25 \).

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Key Concepts

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

Binomial Multiplication
Let's dive into the concept of binomial multiplication. Binomials are algebraic expressions that contain two terms. For example, (4x-5) and (4x+5) are both binomials. When we multiply two binomials, there are specific steps to follow.
Key Points:
  • Binomial multiplication often involves applying special algebraic identities, like the difference of squares formula.
  • Recognize when two binomials follow the form (a - b)(a + b), which simplifies our multiplication process.
Remember, identifying the binomials and knowing the formulas makes solving the problem much easier!
Algebraic Expressions
To fully understand binomial multiplication, let's revisit what algebraic expressions are. An algebraic expression is a combination of numbers, variables, and arithmetic operations such as addition, subtraction, multiplication, and division. In our example, (4x - 5) and (4x + 5) are both algebraic expressions.
Key Features:
  • Terms: Individual parts of the expression separated by plus (+) or minus (-) signs.
  • Coefficients: The number part multiplying the variable, such as 4 in 4x.
  • Variables: Letters that represent unknown values, like x.
Understanding these will help simplify and solve the problems involving algebraic expressions.
Polynomial Simplification
The final step in working with binomials and algebraic expressions often involves simplifying them into a cleaner form, known as polynomial simplification. Simplification makes the expressions easier to understand and work with.
Simplifying Steps:
  • Apply the difference of squares formula: Remember that (a - b)(a + b) simplifies to (a^2 - b^2).
  • Carry out the arithmetic operations: In our example, find (4x)^2 and 5^2, then subtract the results.
  • Combine like terms: Ensure all similar terms are combined into a single expression. In our case, 16x^2 - 25 is the simplified polynomial.
By practicing these steps, polynomial simplification will become second nature, making algebra a breeze!

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