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Chapter 5: Problem 119

Factor each polynomial by grouping. $$ a x-2 a+4 x-8 $$

Short Answer

Expert verified
(x - 2)(a + 4)

Step by step solution

01

Group the terms

Group the terms in pairs to find common factors. Group the polynomial as (ax - 2a) + (4x - 8).
02

Factor out the greatest common factor (GCF) from each pair

Factor out the GCF from each pair separately. For the first pair (ax - 2a), the GCF is a, and for the second pair (4x - 8), the GCF is 4. This gives us: a(x - 2) + 4(x - 2).
03

Factor out the common binomial factor

Notice that (x - 2) is a common factor in both terms. Factor out the common binomial factor to get: (x - 2)(a + 4).

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Key Concepts

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

polynomial factorization
Polynomial factorization is a process used to break down a polynomial into simpler components, or factors, that when multiplied together give back the original polynomial. Think of it as a way to reverse-engineer the expression.
Getting the hang of polynomial factorization can make solving more complex algebraic problems much easier.
In our exercise, the polynomial is expressed as: $$ ax - 2a + 4x - 8.$$

grouping method
One of the methods to factorize polynomials is the grouping method.
This technique is especially useful when dealing with polynomials that have four or more terms. To use the grouping method:
  • Start by dividing the polynomial into groups that have something in common.
  • For our example, we group the terms as follows:$$(ax - 2a) + (4x - 8).$$
  • This allows us to see patterns or common factors within these groups.

Here, we notice that in the first group (ax - 2a), both terms share the variable 'a', and in the second group (4x - 8), both terms share the number 4.
By factoring out these common components, we can consolidate the groups further.
greatest common factor
The greatest common factor (GCF) is the largest factor that divides two or more terms. Finding the GCF is a key step in polynomial factorization as it can simplify the expression.
In our example:
  • For the group (ax - 2a), the GCF is 'a'.
  • For the group (4x - 8), the GCF is 4.

We factor out these GCFs from each group:
$$ a(x - 2) + 4(x - 2). $$
Notice how both groups now contain the identical binomial factor $$(x - 2).$$
Finally, we factor out this common binomial factor:
$$ (x - 2)(a + 4).$$
This gives us the fully-factorized polynomial. Understanding this method is crucial for solving more convoluted expressions efficiently.

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