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Chapter 5: Problem 91

Factor. Assume that variables used as exponents represent positive integers. $$ x^{2 n}-25 $$

Short Answer

Expert verified
The expression is factored as \((x^n + 5)(x^n - 5)\).

Step by step solution

01

Recognize Form

The expression \( x^{2n} - 25 \) is a difference of squares. It can be written in the form \( a^2 - b^2 \), where \( a = x^n \) and \( b = 5 \).
02

Apply Difference of Squares Formula

Use the difference of squares formula, \( a^2 - b^2 = (a + b)(a - b) \), to factor the expression. Substitute \( a = x^n \) and \( b = 5 \) to get \( (x^n + 5)(x^n - 5) \).
03

Verify the Solution

Multiply \( (x^n + 5)(x^n - 5) \) to check that it equals the original expression. Expanding gives \( (x^n)(x^n) - (x^n)(5) + (5)(x^n) - (5)(5) = x^{2n} - 25 \), confirming the factorization.

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Key Concepts

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

Factoring Polynomials
Factoring polynomials is a critical skill in algebra. A polynomial is an expression made up of variables and coefficients, involving addition, subtraction, multiplication, and non-negative integer exponents of variables. When we factor a polynomial, we write it as a product of simpler polynomials. This process is known as factoring, and it can simplify solving equations and understanding the behavior of graphs.

Let鈥檚 look at the exercise: we have to factor the polynomial \( x^{2n} - 25 \).
  • Firstly, notice that \( x^{2n} \) and \( 25 \) are both perfect squares. This sets the stage for using the difference of squares technique.
  • By recognizing that it fits the pattern \( a^2 - b^2 \), where \( a = x^n \) and \( b = 5 \), we can apply the formula \( a^2 - b^2 = (a + b)(a - b) \).
  • The expression \( x^{2n} - 25 \) when factored becomes \((x^n + 5)(x^n - 5)\).
Factoring polynomials like these helps break down complex expressions into manageable parts, making them easier to handle in equations and inequalities.
Exponents in Algebra
Exponents are an essential part of algebra that denote how many times a number, called the base, is multiplied by itself. In the context of our problem, exponents help define the scale of powers for the variable. Let's break it down:
  • An exponent like \( n \) in \( x^n \) signifies that the base \( x \) is multiplied by itself \( n \) times.
  • Therefore, \( x^{2n} \) implies that \( x^n \) itself is raised to the power of 2, resulting in \((x^n)^2\).
  • This multiplication of exponents is governed by exponent rules such as the power of a power rule: \((x^a)^b = x^{a \times b}\).
Understanding exponents helps deal with polynomial expressions, especially when simplifying or factoring, as it indicates the degree and potential complexity of the equation or expression at hand.
Algebraic Identities
Algebraic identities are pre-established formulas that simplify manipulations and transformations of algebraic expressions. They represent patterns that recur in algebra and can be applied universally, like shortcuts in calculation.
The difference of squares identity is incredibly useful:
  • The identity \( a^2 - b^2 = (a + b)(a - b) \) shows how a difference between two squares transforms into a product, which is much simpler to handle.
  • In our exercise, \( x^{2n} - 25 \) is factored into \((x^n + 5)(x^n - 5)\) by applying this identity.
  • This identity not only aids in factoring but also simplifies solving equations by transforming cumbersome expressions into straightforward forms.
Mastering such identities equips students with tools to tackle diverse problems in algebra more efficiently, turning complex algebraic expressions into manageable tasks.

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Most popular questions from this chapter

Multiply. See Section 5.4. $$ (2 x+1)^{2} $$

If \(P(x)=3 x+3, Q(x)=4 x^{2}-6 x+3,\) and \(R(x)=5 x^{2}-7\) find the following. \(3[R(x)]+4[P(x)]\)

Factor each polynomial completely. See Examples 1 through 12. $$ 2 x^{6}+3 x^{3}-9 $$

The volume \(V(x)\) of a box in terms of its height \(x\) is given by the function \(V(x)=x^{3}+2 x^{2}-8 x .\) Factor this expression for \(V(x)\). (GRAPH CANNOT COPY)

Factor. Assume that variables used as exponents represent positive integers. $$ x^{2 n}-36 $$
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