/*! 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 99 Factor each binomial completely.... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 99

Factor each binomial completely. \(8 p^{3}+729 q^{3}\)

Short Answer

Expert verified
(2p + 9q)(4p^2 - 18pq + 81q^2)

Step by step solution

01

- Identify the binomial as a sum of cubes

Recognize that the given binomial is a sum of two cubes: a^3 + b^3Here, identify the cubes: a = 2p and b = 9qbecause a^3 = (2p)^3 = 8p^3 and b^3 = (9q)^3 = 729q^3
02

- Recall the sum of cubes formula

The formula for factoring a sum of cubes is: a^3 + b^3 = (a + b)(a^2 - ab + b^2)
03

- Apply the sum of cubes formula

Substitute the identified terms into the formula: a = 2p and b = 9qa^3 + b^3 = (2p)^3 + (9q)^3 = (2p + 9q)((2p)^2 - (2p)(9q) + (9q)^2)
04

- Simplify the expression

Complete the squares and the product: a^3 + b^3 = (2p + 9q)(4p^2 - 18pq + 81q^2)

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.

Sum of Cubes
Let's first identify the type of expression we're dealing with: a sum of cubes. This means our binomial can be written as a sum of two perfect cubes. In our specific problem, we need to recognize that:

\(8p^3 + 729q^3\) is a sum of cubes because:
- \(8p^3 = (2p)^3 \)
- \(729q^3 = (9q)^3 \)

This insight helps us apply the correct factoring formula for a sum of cubes.
Factoring Techniques
Now that we've identified the binomial as a sum of cubes, we can use specific factoring techniques to break it down.

The primary formula for factoring the sum of cubes is:
\[ a^3 + b^3 = (a + b)(a^2 - ab + b^2) \]

This formula is derived from algebraic identities designed to simplify polynomials of this nature.
By using our identified terms (\(a = 2p\) and \(b = 9q\)), we substitute these into the formula. This substitution converts our complex binomial into a more manageable form.
Polynomial Expressions
Polynomial expressions like the one in our example are algebraic expressions that consist of variables and coefficients. In our case, \(8p^3 + 729q^3\) is a third-degree polynomial. Dealing with polynomials requires techniques such as factoring to simplify or solve them.

In this problem, we aim to break down the polynomial using our sum of cubes formula. Substituting our terms, we get:
\(8p^3 + 729q^3 = (2p)^3 + (9q)^3 = (2p + 9q)\big((2p)^2 - (2p)(9q) + (9q)^2\big)\)

Recognizing polynomial structures and using appropriate formulas are keys to mastering algebra.
Algebraic Identities
Algebraic identities are equations that hold true for any values of the variables involved. These are immensely helpful when factoring expressions like our sum of cubes. In our problem, we applied the sum of cubes identity:
- \[ a^3 + b^3 = (a + b)(a^2 - ab + b^2) \]

With \(a = 2p\) and \(b = 9q\), we replace them to simplify the original expression. Upon simplifying: \((2p)^2 - (2p)(9q) + (9q)^2\) becomes:
- \(4p^2 - 18pq + 81q^2\)

Therefore, \(2p + 9q)\times(4p^2 - 18pq + 81q^2)\). These identities are powerful tools for breaking down complex algebraic expressions.

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 each binomial completely. \(27 r^{3}+1000 s^{3}\)

Solve each problem. Find two consecutive odd integers such that five times their sum is 23 less than their product.

Factor each binomial completely. \(x^{9}+y^{9}\)

Solve each problem. The product of the page numbers on two facing pages of a book is 600 . Find the page numbers.

Solve each problem. Find three consecutive odd integers such that 3 times the sum of all three is 18 more than the product of the first and second integers.
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Applied Mathematics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Logic and Functions

Read Explanation

Pure Maths

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.