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Chapter 0: Problem 20

Factor the difference of two squares. $$ x^{2}-4 $$

Short Answer

Expert verified
(x + 2)(x - 2)

Step by step solution

01

Identify the Difference of Squares

Recognize that the given expression is in the form of a difference of squares. A difference of squares follows the formula: \[a^2 - b^2 = (a + b)(a - b)\]
02

Rewrite the Expression in Terms of Squares

Identify the squares in the given expression. Here, \(x^2\) is a square, and 4 can be rewritten as \(2^2\). Therefore, the expression becomes: \[x^2 - 2^2\]
03

Apply the Difference of Squares Formula

Now that the expression is recognized as a difference of squares, apply the formula: \[x^2 - 2^2 = (x + 2)(x - 2)\]

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Key Concepts

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

Factoring
Factoring is the process of breaking down an expression into simpler components called factors. These factors, when multiplied together, give the original expression. In algebra, factoring helps in simplifying expressions and solving equations. For example, to factor the difference of squares, we identify two terms that are perfect squares being subtracted. Using the formula \(a^2 - b^2 = (a+b)(a-b)\), we can factor expressions like \(x^2 - 4\) into \( (x+2)(x-2) \).
Algebra
Algebra involves the manipulation of mathematical symbols to solve equations and understand relationships. When factoring expressions in algebra, such as the difference of squares, we use specific formulas and techniques. For instance, in the expression \(x^2 - 4\), we recognize that both \(x^2\) and \(4\) (or \(2^2\)) are squares. This recognition allows us to use the difference of squares formula. Algebraic factoring can simplify complex expressions and is a vital skill in solving quadratic equations and higher-degree polynomials.
Polynomials
Polynomials are expressions that consist of variables and coefficients combined using operations like addition, subtraction, and multiplication. A polynomial can have multiple terms, such as \(3x^3 + 2x^2 - x + 5\). When working with polynomials, factoring is a key tool. For the difference of squares, we focus on polynomials that fit the pattern \(a^2 - b^2\). Factoring these polynomials helps to break them down into linear factors, making them easier to solve or simplify. In our example, \(x^2 - 4\) is a polynomial that we can factor into \( (x + 2)(x - 2) \).
Squares
A square in mathematics refers to the result of multiplying a number by itself. In polynomials, a squared term is common, such as \(x^2\) or \(2^2\). The difference of squares involves two perfect squares subtracted from each other, like \(a^2 - b^2\). Recognizing these squares allows us to factor the expression easily. For instance, in \(x^2 - 4\), \(x^2\) is the square of \(x\) and \(4\) is the square of \(2\). Using the difference of squares formula, we can factor the expression into \( (x + 2)(x - 2) \). Understanding squares and their properties is crucial for recognizing and factoring such expressions in algebra.

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Most popular questions from this chapter

Expressions that occur in calculus are given. Factor each expression. Express your answer so that only positive exponents occur. $$6 x^{1 / 2}(2 x+3)+x^{3 / 2} \cdot 8 \quad x \geq 0$$

Simplify each expression. Express your answer so that only positive exponents occur. Assume that the variables are positive. $$ \frac{\left(4 x^{-1} y^{1 / 3}\right)^{3 / 2}}{(x y)^{3 / 2}} $$

Rationalize the denominator of each expression. Assume that all variables are positive when they appear. $$\frac{2}{\sqrt{3}}$$

Simplify each expression. Assume that all variables are positive when they appear. $$-\sqrt{48}+5 \sqrt{12}$$

Simplify each expression. Assume that all variables are positive when they appear. $$(\sqrt[3]{3} \sqrt{10})^{4}$$
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