/*! 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 41 Write in factored form by factor... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 6: Problem 41

Write in factored form by factoring out the greatest common factor. \(12 x^{3}+6 x^{2}\)

Short Answer

Expert verified
The factored form is \(6x^{2}(2x+1)\).

Step by step solution

01

Identify the greatest common factor (GCF)

First, find the greatest common factor of the coefficients 12 and 6, which is 6. Then, determine the common factor of the variables. Both terms have a common factor of \(x^{2}\). Thus, the GCF is \(6x^{2}\).
02

Factor out the GCF

Factor \(6x^{2}\) out of each term in the expression. For the first term, \(12x^{3} \, = \, 6x^{2} \, \times \, 2x\). For the second term, \(6x^{2} \, = \,6x^{2} \, \times \, 1\).
03

Write in factored form

Combine the factored terms to write the expression in factored form: \(12x^{3}+6x^{2}=6x^{2}(2x+1)\).

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
In algebra, the Greatest Common Factor (GCF) is the largest factor that divides two or more numbers. First, you identify the greatest common factor of the given coefficients and the variables. For instance, in the expression 12x鲁 + 6x虏, the coefficients are 12 and 6. The greatest common factor of these coefficients is 6. Next, you look at the variables. Both terms have an x component. The highest power of x that is common to both terms is x虏. Therefore, the GCF for this expression is 6x虏.
Factored Form
Factored form in algebra involves rewriting an expression as the product of its factors. Once you've found the GCF, you can use it to factor out from each term in the expression. For the expression 12x鲁 + 6x虏, you've identified 6x虏 as the GCF. Factoring out 6x虏, you rewrite 12x鲁 as 6x虏 * 2x, and 6x虏 as 6x虏 * 1. This gives a factored form: 12x鲁 + 6x虏 = 6x虏 (2x + 1). Factored form makes expressions simpler to work with, especially in solving equations or simplifying algebraic fractions.
Algebraic Expressions
An algebraic expression is a combination of numbers, variables, and mathematical operations. Expressions like 12x鲁 + 6x虏 contain terms which are made up of coefficients (numbers) and variables (letters). Each term in an expression is separated by a plus (+) or minus (-) sign. The goal of factoring is to simplify these expressions by breaking them down into simpler 'factors' 鈥� elements that, when multiplied together, give the original expression. This process is essential in many areas of algebra, such as simplifying complex equations and finding solutions to algebraic problems.

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 by grouping. \(10 t^{3}-2 t^{2} s^{2}-5 t s+s^{3}\)

Factor completely. If the polynomial cannot be factored, write prime. $$ d^{2}-4 d-45 $$

Factor each polynomial. $$ k^{7}-2 k^{6} m-15 k^{5} m^{2} $$

If a trinomial has a negative coefficient for the squared term, as in \(-2 x^{2}+11 x-12,\) it is usually easier to factor by first factoring out the common factor \(-1 .\) $$ \begin{aligned} -2 x^{2}+11 x-12 \\ =&-1\left(2 x^{2}-11 x+12\right) \\ =&-1(2 x-3)(x-4) \end{aligned} $$ Use this method to factor each trinomial. See Example 7(b). $$ $$ -2 a^{2}-5 a b-2 b^{2} $$

Factor each trinomial completely. $$ 16 x^{2}-40 x+25 $$
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Applied Mathematics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Statistics

Read Explanation

Calculus

Read Explanation

Logic and Functions

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.