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Chapter 11: Problem 54

Multiply and simplify. $$(x+8)(x-8)$$

Short Answer

Expert verified
x^2 - 64

Step by step solution

01

Recognize the Pattern

Identify that (x+8)(x-8) represents a difference of squares formula, which is (a+b)(a-b) = a^2 - b^2.
02

Apply the Difference of Squares Formula

Using the difference of squares formula with a = x and b = 8, calculate a^2 - b^2 which gives x^2 - 64 .
03

Write the Simplified Answer

Combine the results to get the final simplified expression: x^2 - 64.

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Key Concepts

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

algebraic expressions
An algebraic expression is a combination of variables, numbers, and at least one arithmetic operation (such as addition or multiplication). In the given exercise, we have \( (x+8)(x-8) \), which is an algebraic expression involving the variable \( x \) and the numbers \(8\).

The structure of algebraic expressions can vary, but they often include:
  • Constants (like 8)
  • Variables (like x)
  • Operators (like + and -)

Recognizing patterns in algebraic expressions helps in simplifying them. For example, noticing that \( (x+8)(x-8) \) fits the form of the difference of squares formula is key to solving the given problem efficiently.
polynomial multiplication
Polynomial multiplication involves multiplying two or more polynomials together. In this case, the polynomials \( (x+8) \) and \( (x-8) \) need to be multiplied.

The first step is to distribute each term in the first polynomial to each term in the second polynomial. However, recognizing special patterns, like the difference of squares, can simplify this process.

The difference of squares pattern states that \( (a+b)(a-b) = a^2 - b^2 \). Identifying \( a = x \) and \( b = 8 \), we immediately see that multiplying \( (x+8)(x-8) \) yields \( x^2 - 64 \). This pattern saves us from performing each individual multiplication and addition step.
simplification
Once the algebraic expressions are multiplied, the next step is simplification. Simplification means reducing the expression to its most compact and understandable form.

For our example, after applying the difference of squares formula to \( (x+8)(x-8) \), we arrive at \( x^2 - 64 \). There are no like terms to combine further, so this is our simplified expression.

When simplifying, always:
  • Look for patterns (like the difference of squares or perfect square trinomials)
  • Combine like terms
  • Factor where possible
These steps help streamline the process and ensure the final expression is as simple and clear as possible.

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Most popular questions from this chapter

In Exercises \(1-64\), solve each of the given equations. If the equation is quadratic, use the factoring or square root method. If the equation has no real solutions, say so. $$x+\frac{1}{x}=2$$

In Exercises \(75-84\), round your answer to the nearest tenth where necessary. The width of a rectangle is 5 less than twice the length. If the area of the rectangle is 42 sq fit, find the dimensions of the rectangle.

In Exercises \(75-84\), round your answer to the nearest tenth where necessary. Find the length of the diagonal of a square whose side is \(15 \mathrm{cm}\)

Multiply and simplify. $$(x+p)^{2}$$

In Exercises \(85-94\), solve the equations using the square root method. Round off your answers to the nearest hundredth. $$6 w^{2}=1.6$$
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