/*! 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 13 For Problems \(11-46\), factor e... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 6: Problem 13

For Problems \(11-46\), factor each polynomial completely. (Objective 2) $$ 14 x y-21 y $$

Short Answer

Expert verified
The factored form is \(7y(2x - 3)\).

Step by step solution

01

Identify Common Factors

Look at the terms in the polynomial, which are \(14xy\) and \(21y\). Identify the greatest common factor of both terms. Both terms have \(y\) as a common factor and the number 7 is the largest number that divides both 14 and 21.
02

Factor Out the Greatest Common Factor

Factor out the greatest common factor that we identified in Step 1 from each term. The greatest common factor is \(7y\). So, we can rewrite the expression as \(7y(2x - 3)\).
03

Verify the Factored Expression

Check the factored expression \(7y(2x - 3)\) by redistributing the terms to ensure it matches the original polynomial \(14xy - 21y\). Multiply \(7y\) by each term in \(2x - 3\): \(7y \cdot 2x = 14xy\) and \(7y \cdot (-3) = -21y\). This confirms that the factoring is correct.

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.

Greatest Common Factor
The greatest common factor (GCF) is a key concept in factoring polynomials. It's the largest factor that divides each term within the polynomial expression. Understanding the GCF simplifies expressions and makes complex problems more manageable.
To find the GCF of a polynomial, follow these steps:
  • Identify the numerical factors of each term in the polynomial. For example, for the terms 14 and 21, the common numerical factor is 7.
  • Look for any common variables. In the expression \(14xy - 21y\), the variable \(y\) is present in both terms.
  • Multiply these common factors together to find the GCF. Here, the GCF for \(14xy - 21y\) is \(7y\).
Once you have the GCF, factor it out from each term of the polynomial. This process reduces the expression and helps in solving algebraic problems efficiently.
Algebraic Expressions
An algebraic expression is a mathematical phrase representing numbers, variables, and operations. They play an essential role in mathematics education, enabling students to understand and solve various problems. In the expression \(14xy - 21y\), it combines numbers and variables linked by addition or subtraction.
  • Numbers in the expression denote coefficients. In our example, 14 and 21 are coefficients of the terms.
  • Letters like \(x\) and \(y\) represent variables. These symbols can change values depending on the context.
  • Operations such as addition and subtraction connect different parts of the expression.
Factoring transforms these expressions into a simpler or alternative form, shedding light on underlying relationships between terms. It also aids in solving equations by isolating and simplifying parts of the expression.
Mathematics Education
Mathematics education focuses on building foundational skills like factoring to reframe and solve polynomials. Through step-by-step processes, learners develop a deeper understanding of algebraic techniques.
A strong grasp of factoring prepares students for more advanced topics:
  • Understanding the properties of numbers and operations enhances problem-solving abilities.
  • The process is methodical, encouraging logical thinking and attention to detail.
  • Students learn to work through each part of an equation systematically.
Factoring not only applies to algebra but also enhances skills utilized in other areas of math, such as calculus and number theory. By mastering these foundational skills, students cultivate a robust mathematical mindset that serves them in diverse academic and real-world scenarios.

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

$$ \text { For Problems } 41-70 \text {, solve each equation. (Objective } 3 \text { ) } $$ $$ 4 x^{3}-36 x=0 $$

For Problems \(71-88\), set up an equation and solve each problem. (Objective 4) Suppose that the combined area of two squares is 360 square feet. Each side of the larger square is three times as long as a side of the smaller square. How big is each square?

\(x(x-12)=-35\)

For Problems \(71-88\), set up an equation and solve each problem. (Objective 4) Suppose that the length of a certain rectangle is three times its width. If the length is increased by 2 inches, and the width increased by 1 inch, the newly formed rectangle has an area of 70 square inches. Find the length and width of the original rectangle.

Find four consecutive integers such that the product of the two larger integers is 22 less than twice the product of the two smaller integers.
See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Probability and Statistics

Read Explanation

Calculus

Read Explanation

Mechanics Maths

Read Explanation

Applied Mathematics

Read Explanation

Geometry

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.