/*! 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 54 Factor. $$12 a^{3}-12 a^{2}+3 ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 10: Problem 54

Factor. $$12 a^{3}-12 a^{2}+3 a$$

Short Answer

Expert verified
The factored form of the given expression \(12 a^{3}-12 a^{2}+3 a\) is \(a(12a^{2} - 12a + 3)\)

Step by step solution

01

Identifying the common factor

Find the common factor among all the terms in the expression. It can be noticed that each term in the expression \(12 a^{3}-12 a^{2}+3 a\) has 'a' as a common factor.
02

Factoring out the common factor

Factor out the common factor 'a' from each term of the expression. Doing so gives us: \(a(12a^{2} - 12a + 3)\)
03

Simplifying the Expression

Since the expression inside the parenthesis can't be further factored, the final factored form of the given expression is \(a(12a^{2} - 12a + 3)\)

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.

Common Factor
When simplifying expressions, identifying the common factor is a crucial step. A common factor is a number or variable that divides each term in an algebraic expression without leaving a remainder. By factoring out the common factor, we make the expression simpler and more manageable.
  • Firstly, look at each term in the expression to determine the numbers and variables they share.
  • In the expression given in the exercise, each term i.e., \(12a^{3}-12a^{2}+3a\), has a common factor of 'a'.
Factoring out the common factor simplifies the process of handling polynomials, especially when solving equations or reducing fractions. This step is essential, as it lays the groundwork for further simplification or solving.
Polynomial
A polynomial is a mathematical expression consisting of variables, coefficients, and operations of addition, subtraction, and multiplication. Polynomials can have one or more terms, and the degree of a polynomial is determined by the exponent of its highest power.
  • The example from the exercise, \(12a^{3}-12a^{2}+3a\), is a polynomial with three terms, often referred to as a trinomial.
  • The degree of this polynomial is determined by the largest exponent of 'a', which in this case is 3, making it a cubic polynomial.
Once a polynomial is presented, it can be manipulated and factored to find its simplest form. Recognizing the polynomial鈥檚 degree and its terms help in identifying the appropriate factoring techniques.
Factored Form
Factoring is the mathematical process of breaking down an expression into simpler "factors" that, when multiplied together, return the original expression. The factored form is usually more concise and easier to work with, especially for solving polynomial equations.
  • In the original exercise, after factoring out the common 'a', the expression becomes \(a(12a^{2} - 12a + 3)\).
  • This expression is now in its factored form, containing a single variable factor and a simpler polynomial expression.
The factored form is often used to identify roots of polynomials or to simplify complex expressions, making them easier to evaluate or integrate in calculus. Factoring is an essential skill in algebra, and mastering it helps in understanding further mathematical concepts and problem-solving.

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

For Exercises 78 to \(131,\) factor completely. $$4 x^{3}+8 x^{2} y-12 x y^{2}$$

Factor by grouping. $$28+3 z-z^{2}$$

Factor. $$a^{3} b-24 a b-2 a^{2} b$$

Factor by grouping. $$4 y^{2}-11 y z+6 z^{2}$$

Factor by grouping. $$15 b^{2}-115 b+70$$
See all solutions

Recommended explanations on Math Textbooks

Logic and Functions

Read Explanation

Mechanics Maths

Read Explanation

Geometry

Read Explanation

Discrete Mathematics

Read Explanation

Decision Maths

Read Explanation

Pure 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.