Problem 478 Factor the greatest common facto... [FREE SOLUTION]

91影视

Prealgebra

Lynn Marecek, MaryAnne Anthony-Smith

$Math Studyset 91影视 Explanations$ Math

2016 Edition

Chapter 10: Problem 478

Factor the greatest common factor from each polynomial. $$3 a^{4}-24 a^{3}+18 a^{2}$$

Short Answer

Expert verified

\[3a^2(a^2 - 8a + 6)\]

Step by step solution

Identify the Greatest Common Factor (GCF)

Examine each term in the polynomial: - The terms are: $3a^4, -24a^3, 18a^2$.- The numerical part of the terms are 3, -24, and 18. The greatest common factor (GCF) of these numbers is 3.- The variable part has the smallest power as $a^2$. Therefore, the GCF is $3a^2$.

Factor Out the GCF

Divide each term by the GCF ($3a^2$): $3a^4 \rightarrow 3a^2(a^2)$$-24a^3 \rightarrow -24a^3 / 3a^2 = -8a$$18a^2 \rightarrow 18a^2 / 3a^2 = 6$. This can be rewritten as: \[3a^2(a^2 - 8a + 6)\]

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

When solving polynomial factoring problems, one of the first steps is finding the Greatest Common Factor (GCF). The GCF is the largest factor that divides each term in the polynomial without a remainder. It's crucial because factoring out the GCF simplifies the polynomial, making further operations easier.

Here's how you can find the GCF:

Identify the coefficients (numerical parts) of each term and find their GCF. In this case, the coefficients are 3, -24, and 18. The GCF here is 3.
Next, look at the variable parts. The GCF for the variables is determined by the lowest power of the common variable. For our problem, the lowest power of 'a' is $a^2$.

By combining both parts, the GCF for the given polynomial $3a^4-24a^3+18a^2$ is $3a^2$.

Recognizing and factoring out the GCF simplifies the polynomial, setting the stage for further factoring or solving.

Factoring

Factoring is the process of breaking down a complex expression into simpler ones that can be multiplied together to get the original expression. After finding the Greatest Common Factor (GCF), the next step is to factor it out.

For our polynomial $3a^4-24a^3+18a^2$:

We identified the GCF as $3a^2$.
Next, divide each term in the polynomial by $3a^2$:

$3a^4/(3a^2) = a^2$
$-24a^3/(3a^2) = -8a$
$18a^2/(3a^2) = 6$

The result is $a^2 - 8a + 6$. When we factor out the GCF, the polynomial becomes: \[3a^2(a^2 - 8a + 6)\].

Factoring is a valuable skill because it simplifies expressions, making them easier to handle in more complex operations or further factorizations.

Polynomials

Polynomials are expressions consisting of variables and coefficients, combined using addition, subtraction, and multiplication. A polynomial can have constants, single variables, and variables raised to various powers. The degree of the polynomial is determined by the highest power of the variable in the expression.

Understanding polynomials is essential because they're foundational in algebra and calculus. Here are a few key points about polynomials:

Each part of the polynomial (e.g., $3a^4$) is called a 'term'.
The coefficient is the numerical part (here, 3) and it can be any real number.
The variable has an exponent which indicates the degree of the term (here, 4).
The degree of the polynomial $3a^4-24a^3+18a^2$ is 4, because of the highest exponent $a^4$.

Learning to work with polynomials, such as by factoring, opens the door to understanding more complex mathematical concepts.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

91影视

Factor the greatest common factor from each polynomial. $$3 a^{4}-24 a^{3}+18 a^{2}$$

Short Answer

Step by step solution

Identify the Greatest Common Factor (GCF)

Factor Out the GCF

Key Concepts

Greatest Common Factor

Factoring

Polynomials

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Logic and Functions

Discrete Mathematics

Pure Maths

Probability and Statistics

Calculus

Theoretical and Mathematical Physics

Study anywhere. Anytime. Across all devices.