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Chapter 5: Problem 118

Factor each polynomial by grouping. $$ w b+3 w+12+4 b $$

Short Answer

Expert verified
(b + 3)(w + 4)

Step by step solution

01

Group the terms

Group the polynomial into two pairs: (wb + 3w) and (12 + 4b).
02

Factor out the common factor in each group

From the first group (wb + 3w), factor out w: w(b + 3). From the second group (12 + 4b), factor out 4: 4(3 + b).
03

Recognize the common binomial factor

Notice that (b + 3) and (3 + b) are equivalent, so we can write both groups with a common factor: w(b + 3) + 4(b + 3).
04

Factor out the common binomial factor

Factor out the common binomial factor (b + 3) from both terms: (b + 3)(w + 4).

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Key Concepts

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

Polynomial Grouping
When tasked with factoring a polynomial by grouping, the first step is to split the polynomial into pairs of terms. This technique is vital in transforming complex polynomials into simpler, more manageable ones. For instance, take the polynomial:
  • w b + 3 w + 12 + 4 b

To group the terms, reframe it as:
  • (wb + 3w) and (12 + 4b)

By using grouping, we make it easier to identify and factor out common elements, thereby simplifying the entire polynomial expression.
Common Factor
The common factor is an essential concept in factoring polynomials. Identifying and factoring out common factors from groups of terms simplifies the polynomial significantly. For example, in our polynomial:
  • (wb + 3w) and (12 + 4b)
From the first group (wb + 3w), the common factor is 'w':
  • w(b + 3)

From the second group (12 + 4b), the common factor is '4':
  • 4(3 + b)

At this stage, our polynomial looks like this:
  • w(b + 3) + 4(b + 3)
Identifying and factoring out common factors simplifies our polynomial and prepares it for the final steps of factoring.
Binomial Factor
A binomial factor appears when we have common binomials within the polynomial groups after factoring out their respective common factors. This step is critical to further simplify the polynomial. Using our polynomial:
  • w(b + 3) + 4(b + 3)
Notice that (b + 3) is a common binomial factor in both terms. Recognizing this, we can factor out (b + 3):
  • (b + 3)(w + 4)

By factoring out the common binomial factor, the polynomial is now represented in its simplest form. This final simplification step showcases how recognizing and factoring out binomial factors efficiently reduces complex polynomials.

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Most popular questions from this chapter

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