Problem 120 Factor each polynomial by groupi... [FREE SOLUTION]

Chapter 5: Problem 120

Factor each polynomial by grouping. $$ m a-3 a+8 m-24 $$

Short Answer

Expert verified

(m - 3)(a + 8)

Step by step solution

Group the terms

Group the polynomial in pairs to facilitate factoring. The given polynomial can be grouped as follows: a(m - 3) + 8(m - 3).

Factor out the common binomial

Notice that both groups have a common binomial factor (m - 3)a(m - 3) + 8(m - 3) can be factored as (m - 3)(a + 8).

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Key Concepts

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

Polynomial Grouping

Polynomial grouping is an effective method to factor polynomials, especially when they have four or more terms. This method involves grouping terms that have common factors and then factoring them. In the given exercise, the polynomial is split into two groups: $$m a-3 a$$ and $$8 m-24$$. Grouping helps in making the polynomial easier to factor.

The key steps are:

Identify and group pairs or sets of terms.
Look out for common factors within each group.

For example, the expression $$ma - 3a + 8m - 24$$ is grouped into $$(ma - 3a)$$ and $$(8m - 24)$$. This makes the polynomial easier to manage.

Common Binomials

A common binomial factor is an expression that appears in every term of the grouped polynomial. Finding this common factor simplifies the polynomial significantly.

In our exercise, the groups are $$a(m - 3)$$ and $$8(m - 3)$$. Each group has the binomial factor $$(m - 3)$$. This shared factor allows us to factor the polynomial further.

The idea here is to:

Identify the binomial that is common to both groups.
Factor out this common binomial to simplify the polynomial.

For instance, $$a(m - 3) + 8(m - 3)$$ has $$(m - 3)$$ as a common binomial, making it possible to factor out this part easily.

Factoring by Grouping

Factoring by grouping combines polynomial grouping and the identification of common binomials to simplify polynomials. To illustrate with our example, after grouping and spotting the common binomial, we proceed to factor it out.

From the exercise, after grouping, we have $$a(m - 3) + 8(m - 3)$$. The common binomial $$(m - 3)$$ is factored out: $$ (m - 3)(a + 8). $$

To apply factoring by grouping:

First, group the polynomial into sets of expressions with common factors.
Next, find and factor out the common binomial in each group.
Finally, write the polynomial as the product of the common binomial and the remaining factors.

Once you follow these steps, the polynomial becomes much simpler and easier to handle. This method is invaluable for factoring complex polynomials.

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91影视

Factor each polynomial by grouping. $$ m a-3 a+8 m-24 $$

Short Answer

Step by step solution

Group the terms

Factor out the common binomial

Key Concepts

Polynomial Grouping

Common Binomials

Factoring by Grouping

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