Problem 41 Factor completely. $$ a c+c ... [FREE SOLUTION]

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Intermediate Algebra : Concepts and Applications

Marvin L. Bittinger, David J. Ellenbogen, Barbara L. Johnson

$Math Studyset 91影视 Explanations$ Math

10 Edition

Chapter 5: Problem 41

Factor completely. $$ a c+c d-a b-b d $$

Short Answer

Expert verified

(a + d)(c - b)

Step by step solution

Group the terms

Group the given terms in pairs to make factoring easier. The given expression is: a c+c d-a b-b dSo group them as follows: (a c + c d) -(a b + b d)

Factor out the common factor from each group

Factor out the common factor from each of the grouped terms. In the first group (a c + c d), the common factor is c. In the second group -(a b + b d), the common factor is b. So the expression now looks like this: c(a + d) - b(a + d)

Factor out the common binomial

Now, notice that a + d is a common factor in the expression c(a + d) - b(a + d). Factor a + d out from both terms: (a + d)(c - b)

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Key Concepts

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

Grouping Terms

When factoring polynomials, one helpful strategy is to group the terms. This simplifies the polynomial, making it easier to find common factors later. First, identify pairs of terms that can be grouped together. For the expression$ac + cd - ab - bd$, we can group the terms as follows:

The expression becomes $(ac + cd) - (ab + bd)$.

Notice that we're pairing terms in a way that we'll be able to factor them more easily. Grouping terms is the first step toward breaking down complex expressions into simpler, more manageable parts.

Common Factor

After grouping the terms, the next step is to factor out any common factors within each group. A common factor is a term shared by each term within a group.

In our grouped expression, $ac + cd$ and $ab + bd$, we can see that:

In $ac + cd$, the common factor is $c$.
In $ab + bd$, the common factor is $b$.

So, we factor these common terms out:
\c(a + d) - b(a + d)\.

Finding and factoring out common factors simplifies each part of the expression and sets us up for the final step in the process.

Binomial Factorization

Finally, we look at our simplified expression to see if there are any common binomial factors to be factored out.

In our expression $c(a + d) - b(a + d)$, we can see that $a + d$ appears in both terms. This means $a + d$ is a common binomial factor.

We can factor out $a + d$ from both parts of the expression, giving us:
$ (a + d)(c - b) $.

This completes the factorization of the original polynomial. Binomial factorization often involves recognizing patterns where a common binomial expression can be pulled out, greatly simplifying the original polynomial.

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91影视

Factor completely. $$ a c+c d-a b-b d $$

Short Answer

Step by step solution

Group the terms

Factor out the common factor from each group

Factor out the common binomial

Key Concepts

Grouping Terms

Common Factor

Binomial Factorization

One App. One Place for Learning.

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