Problem 463 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 463

Factor the greatest common factor from each polynomial. $$a b+8 b$$

Short Answer

Expert verified

The factored form is $b(a + 8)$.

Step by step solution

Identify the common factor

Look at the terms in the polynomial: $ab + 8b$. Identify the common factor in each term. In this case, both terms have the factor $b$.

Factor out the common factor

Factor $b$ out from each term in the polynomial. Divide each term by $b$ and write the polynomial as $b(a + 8)$.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Greatest Common Factor (GCF)

When factoring polynomials, the first step is to find the Greatest Common Factor (GCF). The GCF is the largest factor that can divide all terms in the polynomial.

For example, in the polynomial $ ab + 8b $, the common factor shared by both terms is $ b $.
By identifying the GCF, we simplify the expression and make it easier to work with.

Here鈥檚 how to find the GCF of a polynomial:

Break down each term into its factors
Compare the factors
Pick out the largest factor that appears in each term

For $ ab + 8b $:
Factors of $ ab $ are $ a, b $
Factors of $ 8b $ are $ 8, b $

The common factor between both terms is $ b $. Thus, $ b $ is the GCF.

Polynomial

Understanding what a polynomial is can make the factoring process much clearer.
A polynomial is a mathematical expression involving a sum of powers of one or more variables multiplied by coefficients.
Here are some key points:

Each distinct power of the variable is a 'term'
Terms are separated by '+' or '-' signs
The degree of a polynomial is the highest power of the variable

For instance, in our example of $ ab + 8b $,

We have two terms: $ ab $ and $ 8b $
Polynomials can have constants, variables, and coefficients

Understanding that a polynomial is a sum of terms helps us focus on each term individually when factoring.

Factoring

Factoring is the process of breaking down a complex expression into simpler components (factors) that, when multiplied together, give back the original expression.

Here is how you can factor a polynomial:

Identify and factor out the GCF from all terms.
Rewrite the polynomial using the common factor.
Simplify the resulting expression if possible.

Let's go through our example: $ ab + 8b $.

First, we recognized that $ b $ is the GCF.
Factoring $ b $ out of each term gives: $ b(a) + b(8) $.
This can be written as $ b(a + 8) $.

By factoring, we transformed the polynomial into a simpler expression: $ b(a + 8) $. Now, it's easier to see the underlying structure of the polynomial.

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91影视

Factor the greatest common factor from each polynomial. $$a b+8 b$$

Short Answer

Step by step solution

Identify the common factor

Factor out the common factor

Key Concepts

Greatest Common Factor (GCF)

Polynomial

Factoring

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