/*! 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 59 Find each product. \((x+y)^{2}... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 59

Find each product. \((x+y)^{2}(x-y)^{2}\)

Short Answer

Expert verified
(x^2 - y^2)^2

Step by step solution

01

- Use the identity for square of binomials

Recall the identity \(a^2 - b^2 = (a + b)(a - b)\). We will leverage this to simplify \( (x + y)^2 (x - y)^2 \).
02

- Apply the identity to the given expression

Recognize that \( (x+y)^2 \) and \( (x-y)^2 \) can be seen as \( a \) and \( b \). Therefore, \( (x + y)^2 (x - y)^2 = [(x + y)(x - y)]^2 \).
03

- Expand the expression

Expand the expression inside the bracket: \( (x + y)(x - y) = x^2 - y^2 \). Thus, \( [(x + y)(x - y)]^2 = (x^2 - y^2)^2 \).
04

- Simplify the final expression

Now square the resulting expression: \[ (x^2 - y^2)^2 = x^4 - 2x^2y^2 + y^4 \].

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.

Algebraic Identities
Algebraic identities are like shortcuts for simplifying and solving equations. They are pre-established equations that hold true for any value of the variables involved.
For instance, one common algebraic identity is the difference of squares: \(a^2 - b^2 = (a + b)(a - b)\).
When you recognize these patterns, it makes your calculations faster and more accurate. In our exercise, we used this identity to simplify the expression \((x + y)^2 (x - y)^2\) by treating each binomial square as parts of our identity.
Polynomial Multiplication
Polynomial multiplication can sometimes seem tricky, but it鈥檚 just about distributing terms and combining like terms.
For multiplying binomials, we use a distributive approach that鈥檚 often remembered as FOIL: First, Outer, Inner, Last. However, with more complex expressions, certain identities or patterns can significantly simplify the process.
In our exercise, instead of directly multiplying \((x + y)^2 (x - y)^2\), we grouped the terms using an identity which ultimately made the multiplication much simpler and led directly to our solution.
Square of Binomials
The square of a binomial is a special multiplication case. When you square a binomial, like \((x + y)^2\), it means multiplying the binomial by itself: \((x + y)\cdot(x + y)\).
The general formula is: \[ (a + b)^2 = a^2 + 2ab + b^2 \].
A similar one exists for subtraction: \[ (a - b)^2 = a^2 - 2ab + b^2 \].
Recognizing these can speed up calculations and reduce error. In the exercise, squaring \((x + y)\) and \((x - y)\) separately helped lay groundwork for using other identities.
Difference of Squares
This saved multiple steps of individual term multiplication and made the process clearer.

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

Use scientific notation to calculate the answer to each problem. Astronomers using the Spitzer Space Telescope discovered a twisted double- helix nebula, a conglomeration of dust and gas stretching across the center of the Milky Way galaxy. This nebula is \(25,000\) light-years from Earth. If one light-year is about \(6,000,000,000,000\) (that is, 6 trillion) miles, about how many miles is the twisted double-helix nebula from Earth?

Simplify by writing each expression wth positive exponents. Assume that all variables represent nonzero real numbers. $$ \frac{\left(3 p^{-2} q^{3}\right)^{2}\left(5 p^{-1} q^{-4}\right)^{-1}}{\left(p^{2} q^{-2}\right)^{-3}} $$

Perform each division. $$ \left(x^{3}+2 x^{2}-3\right) \div(x-1) $$

Subtract. \(5 t^{2}+2 t-6\) \(5 t^{2}-3 t-9\) \(\hline\)

Simplify: $$ -8(-3 x+7)-4(2 x+3) $$
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Decision Maths

Read Explanation

Discrete Mathematics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Statistics

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.