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Chapter 0: Problem 24

Factor the difference of two squares. $$ x^{2}-25 $$

Short Answer

Expert verified
The factored form of \(x^2 - 25\) is \((x + 5)(x - 5)\).

Step by step solution

01

Identify the terms

The given expression is \(x^2 - 25\). Recognize that this is a binomial (two terms) which fits the form of the difference of two squares, \(a^2 - b^2\).
02

Rewrite each term as a square

Express each term as a square: \(x^2\) is already a square term, and \(25\) can be written as \(5^2\). So the expression becomes \(x^2 - 5^2\).
03

Apply the difference of squares formula

Use the formula for the difference of squares: \(a^2 - b^2 = (a + b)(a - b)\). In this case, \(a = x\) and \(b = 5\).
04

Substitute and factorize

Substitute the values of \(a\) and \(b\) into the formula: \[(x + 5)(x - 5)\]. Therefore, the factored form of \(x^2 - 25\) is \((x + 5)(x - 5)\).

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Key Concepts

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

Factoring Polynomials
To start, let's talk about factoring polynomials. Polynomials are algebraic expressions made up of terms connected by addition or subtraction. When we factor a polynomial, we essentially rewrite it as a product of simpler expressions. This makes it easier to solve equations or simplify expressions.

In the given exercise, we have the polynomial expression:

\[ x^2 - 25 \]

Here, we're looking at a specific type of factoring known as the difference of squares. The goal is to recognize and apply patterns to rewrite the polynomial as a product of simpler terms.
Algebraic Identities
Algebraic identities are equations that are true for all values of the variables involved. These identities help simplify complex expressions and solve equations. One important identity used in this exercise is the difference of squares:

\[ a^2 - b^2 = (a + b)(a - b) \]

This formula states that the difference between two squared terms can be expressed as the product of the sum and difference of the terms being squared.

In our exercise, we rewrite 25 as \(5^2\) making the expression:

\[ x^2 - 5^2 \]

By applying the difference of squares identity, we then rewrite this as:

\[ (x + 5)(x - 5) \]

This way, we've factored the original expression.
Binomial Expressions
Binomial expressions are polynomials with exactly two terms, like the one in the example:

\[ x^2 - 25 \]

In the context of the difference of squares, we're often working with binomials because they involve two squared terms subtracted from each other. By recognizing the binomial form, we apply the difference of squares identity to simplify the expression.

So, anytime you see a binomial expression resembling \(a^2 - b^2\), remember you can convert it into two binomials multiplied together:

\[ (a + b)(a - b) \]

This is a powerful tool for breaking down and simplifying algebraic expressions.

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Most popular questions from this chapter

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

Expressions that occur in calculus are given. Factor each expression. Express your answer so that only positive exponents occur. $$3 x^{-1 / 2}+\frac{3}{2} x^{1 / 2} \quad x>0$$

Simplify each expression. Assume that all variables are positive when they appear. $$(5 \sqrt{8})(-3 \sqrt{3})$$

Rationalize the denominator of each expression. Assume that all variables are positive when they appear. $$\frac{\sqrt{3}}{5-\sqrt{2}}$$

Simplify each expression. $$100^{3 / 2}$$
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