/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 35 In Exercises \(31-40,\) factor t... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 0: Problem 35

In Exercises \(31-40,\) factor the difference of two squares. $$9 x^{2}-25 y^{2}$$

Short Answer

Expert verified
The factored form of the expression \(9x^{2} - 25y^{2}\) is \((3x - 5y)(3x + 5y)\).

Step by step solution

01

Identify a and b

In the expression \(9x^{2} - 25y^{2}\), identify a and b from the formula \(a^{2} - b^{2}\). In this case, \(a^{2}\) is \(9x^{2}\) which gives us that \(a = 3x\), and \(b^{2}\) is \(25y^{2}\), which gives us that \(b = 5y\).
02

Use the difference of two squares formula

Substitute a and b into the formula \((a - b)(a + b)\) to factorize the expression. This gives us \((3x - 5y)(3x + 5y)\).
03

Write the final solution

The factored form of the given expression \(9x^{2} - 25y^{2}\) is \((3x - 5y)(3x + 5y)\). Write down this as the final answer.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Factoring Expressions
Factoring expressions is a fundamental skill in algebra that involves breaking down a composite mathematical expression into a product of simpler factors. This makes it easier to evaluate or simplify expressions and solve equations.

A common scenario in algebra is recognizing specific forms like the difference of squares. The expression \(9x^2 - 25y^2\) can be identified as a difference of two squares because both terms separately form perfect squares.
  • \(9x^2\) is \((3x)^2\)
  • \(25y^2\) is \((5y)^2\)
Once these squares are identified, they guide us into using the difference of squares formula to factor the expression efficiently.
Polynomial Identities
Polynomial identities provide a bridge to understanding how different polynomials behave under various operations, like addition, subtraction, and multiplication. One key identity that emerges often is the difference of squares identity.

When we write \(a^2 - b^2\), we recognize that it can directly be transformed using the identity:
\[(a - b)(a + b)\]
This is the basis for the factoring process observed in the expression \(9x^2 - 25y^2\).
  • Identifying \(a = 3x\) and \(b = 5y\) allows us to utilize this identity directly.
  • The expression simplifies and factors into two linear binomials, \((3x - 5y)(3x + 5y)\).
These identities simplify complex expressions and make them easier to work with, supporting both algebraic manipulation and problem-solving.
Algebraic Formulas
Algebraic formulas like the difference of squares are key tools in simplifying and solving algebraic expressions and equations. These formulas are precise instructions for transforming one expression into another in a way that is mathematically valid and useful.

The specific formula for the difference of squares is:\[a^2 - b^2 = (a - b)(a + b)\]This formula works because it represents the geometric difference of the two square areas, allowing intercepts that simplify expressions easily.
  • This helps in breaking down cubic or quartic equations into more manageable forms.
  • It plays a crucial role not only for factoring but also in roots finding.
Understanding and applying algebraic formulas effectively equips students with the skills to tackle a range of mathematical problems, simplifying tasks and reducing potential arithmetic errors.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Write each number in decimal notation. $$ 7.86 \times 10^{-4} $$

In Exercises \(1-10\), factor out the greatest common factor. $$x^{2}(2 x+5)+17(2 x+5)$$

simplify each algebraic expression. $$ 5(3 x+4)-4 $$

state the name of the property illustrated. $$ (x+4)+[-(x+4)]=0 $$

Simplify each exponential expression $$ \frac{8 x^{20}}{2 x^{4}} $$
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Statistics

Read Explanation

Decision Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Geometry

Read Explanation

Calculus

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.