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Chapter 5: Problem 104

Factor. $$ b d x^{2}+a d x+b c x+a c $$

Short Answer

Expert verified
(b x + a)(d x + c)

Step by step solution

01

- Group the Terms

Group the terms in pairs that can be factored. Rewrite the expression as: \[ (b d x^{2} + a d x) + (b c x + a c) \]
02

- Factor Out Common Terms from Each Group

Factor out the greatest common factor from each pair: \[ d x (b x + a) + c (b x + a) \]
03

- Factor Out the Common Binomial

Notice that \( b x + a \) is common in both terms. Factor out \( b x + a \): \[ (b x + a)(d x + c) \]

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Key Concepts

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

Grouping Terms
The first step in factoring polynomials like the one in the exercise is to group the terms. Grouping terms means rearranging them into smaller sets that are easier to manage.
For example, given the polynomial: equation: bdx^{2}+adx+bcx+ac

We can group the terms into pairs like this: equation: (b d x^{2} + a d x) + (b c x + a c)

This arrangement allows us to factor each group separately, simplifying the polynomial into more manageable parts. Grouping is useful when terms contain shared factors that can be factored out more easily from smaller subsets.
Common Factors
After grouping the terms, the next step is to factor out any common factors from each group.
A common factor is a term that is shared among all parts of a group. For our example, the common factors in each group are:
  • In the first group equation: (bdx^{2} + adx), the common factor is dx.
  • In the second group equation: (bcx + ac), the common factor is c.
Factoring these gives us: equation: d x (b x + a) + c (b x + a)

By pulling out the greatest common factor from each group, we simplify the polynomial, reducing the expression to more recognizable forms that can be further factored.
Binomial Factor
In the final step of factoring our polynomial, we focus on identifying and factoring out the common binomial factor.
A binomial is an algebraic expression containing two terms. Notice in our factored expression from the last step: equation: d x (b x + a) + c (b x + a)

The binomial equation: (bx + a) appears in both terms.

We can factor out this common binomial term: equation: (b x + a)(d x + c)

By recognizing and factoring out the binomial factor, we simplify the polynomial down to a product of two binomials.
  • This makes the polynomial easier to handle.
  • It also reveals the underlying structure of the expression.
Understanding these steps allows you to factor more complex polynomials efficiently.

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