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Chapter 6: Problem 83

Multiply. $$ (x+2)(x-2) $$

Short Answer

Expert verified
The solution is \[ x^2 - 4 \].

Step by step solution

01

Apply the Difference of Squares Formula

The given expression \(x+2\)(x-2) is a classic example of the difference of squares formula. Recall that \[ (a+b)(a-b) = a^2 - b^2 \].
02

Identify Variables

In the given expression, identify \(a \) and \(b \) as follows: \(a = x \) and \( b = 2 \).
03

Substitute and Simplify

Use the difference of squares formula to get \(a^2 - b^2 \). Substitute \(a = x \) and \( b = 2 \), resulting in \[ x^2 - 2^2 \]. Simplify this to get \[ x^2 - 4 \].

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Key Concepts

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

algebraic expressions
Algebraic expressions are combinations of variables, numbers, and operations like addition, subtraction, multiplication, and division. They provide a way to generalize arithmetic operations and solve for unknown values.

In the expression \(x+2\)(x-2), we have a polynomial involving the variable \(x\) and constants. Understanding how to manipulate such expressions is crucial in algebra. Breaking them down step by step helps simplify and solve complex problems.

Key points to remember about algebraic expressions:
  • They can represent real-world situations.
  • They contain variables, which are symbols like \(x\) and \(y\) that represent unknown numbers.
  • Constants are fixed values within the expressions.
  • Expressions can have one or more terms.
polynomial multiplication
Polynomial multiplication involves multiplying two or more polynomials together. You might see this when multiplying expressions like \(x + 2\) and \(x - 2\).

When multiplying polynomials, each term in the first polynomial must be multiplied by each term in the second polynomial. It鈥檚 like distributing each term to every other term.

However, certain patterns, like the difference of squares, make polynomial multiplication simpler. In our example, \(x + 2\) and \(x -2\) follow this pattern. Instead of multiplying everything out, you can use the formula \[ (a + b)(a - b) = a^{2} - b^{2} \].

This fast-tracks the process, turning our example into \(x^{2} - 4\) quickly and efficiently.
factoring techniques
Factoring techniques involve breaking down complex polynomials into simpler parts. One common technique is recognizing patterns like the difference of squares.

The difference of squares pattern tells us that \[ (a + b)(a - b) = a^{2} - b^{2} \]. This makes it easier to multiply and factor polynomials. For example, \( (x + 2)(x - 2) = x^{2} - 4 \).

Other factoring techniques include:
  • Finding the greatest common factor (GCF).
  • Factoring by grouping.
  • Using the quadratic formula for more complex polynomials.


Practical tips to improve your factoring skills:
  • Always look for common factors first.
  • Recognize special patterns.
  • Practice with different problems to become more comfortable.

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Most popular questions from this chapter

Write an equivalent expression by factoring out the smallest power of \(x\) in each of the following. $$ x^{3 / 4}+x^{1 / 2}-x^{1 / 4} $$

Factor completely. Remember to look first for a common factor. If a polynomial is prime, state this. $$ 49 a^{4}+100 $$

Determine whether each of the following is a difference of squares. $$ -1+64 t^{2} $$

Factor completely. Remember to look first for a common factor. If a polynomial is prime, state this. $$ q^{2}+1 $$

Solve. Round any irrational solutions to the nearest thousandth. $$ x^{3}+3 x^{2}+x-1=0 $$
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