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Chapter 6: Problem 42

In the following exercises, factor by grouping. $$ p q-10 p+8 q-80 $$

Short Answer

Expert verified
\( (q - 10)(p + 8) \)

Step by step solution

01

Group the terms

Group the terms in pairs: \( (pq - 10p) + (8q - 80) \)
02

Factor out the greatest common factor (GCF) in each group

Factor out the GCF from each of the two groups: \( p(q - 10) + 8(q - 10) \)
03

Factor out the common binomial factor

Notice that \(q - 10\) is a common factor in both groups. Factor \(q - 10\) out: \( (q - 10)(p + 8) \)

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Key Concepts

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

Greatest Common Factor (GCF)
The Greatest Common Factor (GCF) is the largest factor that divides two or more numbers. When factoring by grouping, finding the GCF in each pair makes the process smoother and simpler.
In our problem, we first grouped the terms into pairs: \(pq - 10p\) and \(8q - 80\).
Next, we find the GCF of each group:
  • For the pair \(pq - 10p\), the GCF is \(p\) since both terms share a common factor of \(p\).
  • For the pair \(8q - 80\), the GCF is \(8\) because 8 is the largest number that divides both 8 and 80.
By factoring out these GCFs, we simplify the expression to \(p(q - 10) + 8(q - 10)\).
Polynomial Factoring
Polynomial factoring involves expressing a polynomial as a product of its factors. This makes complex expressions easier to work with in algebraic equations.
In the given exercise, we used a common method: factoring by grouping. This is particularly useful for polynomials with four terms.
First, we grouped the polynomial terms: \(pq - 10p + 8q - 80\) becomes \( (pq - 10p) + (8q - 80) \).
Next, we factored out the GCF from each group, resulting in \( p(q - 10) + 8(q - 10) \).
Finally, we factored out the common binomial factor, \(q - 10\), leaving us with \( (q - 10)(p + 8) \). This completes the polynomial factoring process.
Binomial Factor
A binomial factor is a polynomial with exactly two terms. Recognizing and factoring out binomial factors is crucial in simplifying expressions.
In our example, after grouping and factoring out the GCF, we found that \(q - 10\) appeared in both groups as a binomial factor.
We factored out the common binomial factor: \( p(q - 10) + 8(q - 10) \) becomes \((q - 10)(p + 8)\).
This final step is key to simplifying the polynomial to its factored form. Identifying binomial factors correctly makes solving algebraic equations more manageable.

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