/*! 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 27 Factor each difference of two sq... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 27

Factor each difference of two squares. See Example 2. $$ 36 x^{4} y^{2}-49 z^{6} $$

Short Answer

Expert verified
\((6x^2y - 7z^3)(6x^2y + 7z^3)\)

Step by step solution

01

Identify the Difference of Squares Formula

The difference of squares formula is \(a^2 - b^2 = (a-b)(a+b)\). Our given equation is \(36x^4y^2 - 49z^6\). Recognize that each term is a perfect square.
02

Express Each Term as a Square

Rewrite \(36x^4y^2\) as \((6x^2y)^2\) and \(49z^6\) as \((7z^3)^2\). This will help us apply the difference of squares formula.
03

Apply the Difference of Squares Formula

Use the formula \((a-b)(a+b)\). Substitute \(a = 6x^2y\) and \(b = 7z^3\) into the formula. This gives us \((6x^2y - 7z^3)(6x^2y + 7z^3)\) as the factored form.
04

Write the Final Answer

The fully factored form of the expression \(36x^4y^2 - 49z^6\) is \((6x^2y - 7z^3)(6x^2y + 7z^3)\).

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 Polynomials
Factoring polynomials involves breaking down a complex expression into simpler factors or terms that can be multiplied together to form the original expression. In the context of the difference of squares, this involves a formula that can help simplify expressions quickly. The difference of squares formula is \(a^2 - b^2 = (a-b)(a+b)\). Breaking down each element of the polynomial to fit this structure is the key to simplifying it.

To factor a polynomial:
  • Identify terms that are perfect squares.
  • Express each term as a square.
  • Apply the difference of squares formula.
By understanding the structure of polynomials and how to apply factoring techniques like the difference of squares, often complex algebraic expressions become simpler and more manageable.
Perfect Squares
A perfect square is an expression that results from multiplying a number or a variable by itself. Commonly in algebra, perfect squares appear in the form of \(n^2\), where \(n\) is any term. Recognizing perfect squares in an expression allows for the application of special factoring techniques.

For instance, in the expression \(36x^4y^2\), the term can be rewritten as \((6x^2y)^2\), and similarly, \(49z^6\) can be rewritten as \((7z^3)^2\). This simplifies recognizing these as perfect squares:
  • The coefficient \(36\) is a perfect square because it equals \(6^2\).
  • Similarly, \(49\) is \(7^2\).
  • The powers of the variables are also even, making them perfect squares such as \(x^4 = (x^2)^2\) and \(z^6 = (z^3)^2\).
By identifying terms as perfect squares, you can effectively apply the difference of squares formula to factor polynomials.
Algebraic Expressions
Algebraic expressions are combinations of numbers, variables, and operations (like addition or subtraction) that represent a particular value or set of values. Polynomials like \(36x^4y^2 - 49z^6\) are a type of algebraic expression that can be particularly complex due to the powers of variables involved.

Understanding each component within an algebraic expression helps in breaking it down:
  • Recognize the constants (numbers without variables) and coefficients (numbers paired with variables).
  • Identify like terms, which are terms that share the same variable raised to the same power.
  • Understand the operations connecting these terms, such as subtraction in the case of a difference of squares.
In essence, mastering algebraic expressions by evaluating their structure allows for effective factoring and simplification, turning seemingly complex tasks into manageable ones when tackled step by step.

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

Factor the expression in part a. Then use your answer from part a to give the factorization of the expression in part b. (No new work is necessary!) a. \(a^{6}-b^{3}\) b. \(a^{6}+b^{3}\)

Describe each expression in words:$$ (x+y)^{2} \quad(x-y)^{2} \quad(x+y)(x-y) $$

Calculus. In the advanced mathematics course called Calculus, an important polynomial function is $$f(x)=1+x+\frac{x^{2}}{2}+\frac{x^{3}}{6}+\frac{x^{4}}{24}$$ Refer to the polynomial on the right side of the equation. A. How many terms does the polynomial have? B. What is the degree of the polynomial? C. What is the coefficient of the fourth term? D. Find \(f(1)\)

What are the only integer values of \(b\) for which \(9 m^{2}+b m-1\) can be factored?

$$ \begin{aligned} &f(x)=2 x^{2}-4 x+2\\\ &x=-1,0,1,2,3 \end{aligned} $$ $$ \begin{array}{|r|r|} \hline x & f(x) \\ \hline-1 & \\ 0 & \\ 1 & \\ 2 & \\ 3 & \\ \hline \end{array} $$
See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Discrete Mathematics

Read Explanation

Mechanics Maths

Read Explanation

Decision Maths

Read Explanation

Geometry

Read Explanation

Probability and Statistics

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.