/*! 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 53 Find each product. $$ 3 y\le... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 53

Find each product. $$ 3 y\left(5 y^{3}+2\right)\left(5 y^{3}-2\right) $$

Short Answer

Expert verified
75 y^7 - 12y

Step by step solution

01

Recognize and Apply the Difference of Squares Formula

Observe that the expression \((5 y^3 + 2)(5 y^3 - 2)\) resembles the difference of squares formula \(a^2 - b^2 = (a + b)(a - b)\). Here, \(a = 5y^3\) and \(b = 2\).
02

Simplify using the Difference of Squares Formula

Write the simplified form: \( (5 y^3 + 2)(5 y^3 - 2) = (5 y^3)^2 - 2^2 \). Then calculate the squares: \( (5 y^3)^2 = 25 y^6 \) and \( 2^2 = 4 \). Thus, the expression simplifies to \( 25 y^6 - 4 \).
03

Multiply by 3y

Now multiply the simplified expression by the remaining part \(3y\): \[ 3y \times (25 y^6 - 4) \]. Distribute \(3y\) across both terms: \[ 3y \times 25 y^6 - 3y \times 4 = 75 y^7 - 12y \].

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 expressions
Algebraic expressions are combinations of variables, numbers, and operators (like +, -, *, and /) assembled to represent a value. For example, in the given exercise, the expression is 3y(5y鲁 + 2)(5y鲁 - 2).

This expression includes:
  • Variables: 'y'
  • Numbers: '3', '5', '2'
  • Operators: '*', '+'
Understanding the structure and components of algebraic expressions is essential for simplifying and solving them. This exercise involves recognizing patterns, such as the difference of squares, to simplify complex expressions. Differentiating between terms and factors helps in manipulating the expression correctly. For instance, 5y鲁 + 2 and 5y鲁 - 2 are polynomial factors in this expression.
polynomial multiplication
Polynomial multiplication involves multiplying two or more polynomials to find their product. In this exercise, we are multiplying three parts: 3y, (5y鲁 + 2), and (5y鲁 - 2).

The first steps involve recognizing useful patterns, like the difference of squares, to simplify the problem. The formula for the difference of squares is:
factoring techniques
Factoring techniques are methods used to break down complex expressions into simpler ones that can be easily solved or simplified. In this exercise, the technique of recognizing the difference of squares is applied first.

The difference of squares formula states:

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 standard notation. $$ 2.54 \times 10^{-3} $$

Simplify each expression so that no negative exponents appear in the final result. Assume that all variables represent nonzero real numbers. $$ \left(z^{-4} x^{3}\right)^{-1} $$

Simplify each expression so that no negative exponents appear in the final result. Assume that all variables represent nonzero real numbers. $$ -4 r^{-2}\left(r^{4}\right)^{2} $$

Write each number in standard notation. $$ -6 \times 10^{4} $$

Simplify each expression. Assume that all variables represent nonzero real numbers. $$ \frac{\left(2 m^{2} p^{3}\right)^{2}\left(4 m^{2} p\right)^{-2}}{\left(-3 m p^{4}\right)^{-1}\left(2 m^{3} p^{4}\right)^{3}} $$
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Statistics

Read Explanation

Discrete Mathematics

Read Explanation

Logic and Functions

Read Explanation

Applied Mathematics

Read Explanation

Mechanics Maths

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.