Problem 301
Question
Factor completely. \(15 p q-15 p+12 q-12\)
Step-by-Step Solution
Verified Answer
(q - 1)(15p + 12)
1Step 1: Group the Terms
First, group the terms to make factoring easier. Group the first two terms together and the last two terms together: (15pq - 15p) + (12q - 12)
2Step 2: Factor Out the Greatest Common Factor (GCF) in Each Group
Factor out the greatest common factor from each group. From the first group: 15p(q - 1) From the second group: 12(q - 1) Thus, it becomes: 15p(q - 1) + 12(q - 1)
3Step 3: Factor Out the Common Binomial Factor
Now, factor out the common binomial factor (q - 1): (q - 1) (15p + 12)
Key Concepts
greatest common factorgrouping termsbinomial factorizationelementary algebra
greatest common factor
The 'greatest common factor' (GCF) is the largest number that divides two or more numbers. In algebra, we also look at variables along with numbers. Finding the GCF helps in simplifying expressions and solving equations.
Let's look at the expression given: \(15pq - 15p + 12q - 12\).
We begin by splitting the expression into two groups: \((15pq - 15p) + (12q - 12)\).
Now, for the first group \(15pq - 15p\), we identify that both terms have the factor of \(15p\).
- Factoring out 15p from the first group, we get: \(15p(q - 1)\).
For the second group \(12q - 12\), both terms have the factor 12.
- Factoring out 12 from the second group, we get: \(12(q - 1)\).
This makes our large expression easier to manage!
Let's look at the expression given: \(15pq - 15p + 12q - 12\).
We begin by splitting the expression into two groups: \((15pq - 15p) + (12q - 12)\).
Now, for the first group \(15pq - 15p\), we identify that both terms have the factor of \(15p\).
- Factoring out 15p from the first group, we get: \(15p(q - 1)\).
For the second group \(12q - 12\), both terms have the factor 12.
- Factoring out 12 from the second group, we get: \(12(q - 1)\).
This makes our large expression easier to manage!
grouping terms
Grouping terms is a technique that makes complex expressions simpler to factor. By dividing an expression into smaller groups, we can handle and factor them more easily.
For example, start with the expression \(15pq - 15p + 12q - 12\).
Notice we can split it into two groups: \((15pq - 15p) + (12q - 12)\).
Each group has a common factor that we can factor out:
- In the first group, \(15p\) is common, so we factor it out: \(15p(q - 1)\).
- In the second group, \(12\) is common, so we factor it out: \(12(q - 1)\).
When we rewrite the original expression using these factors, it becomes: \(15p(q - 1) + 12(q - 1)\).
As you can see, this makes it easier to see the common factor and proceed with further simplification!
For example, start with the expression \(15pq - 15p + 12q - 12\).
Notice we can split it into two groups: \((15pq - 15p) + (12q - 12)\).
Each group has a common factor that we can factor out:
- In the first group, \(15p\) is common, so we factor it out: \(15p(q - 1)\).
- In the second group, \(12\) is common, so we factor it out: \(12(q - 1)\).
When we rewrite the original expression using these factors, it becomes: \(15p(q - 1) + 12(q - 1)\).
As you can see, this makes it easier to see the common factor and proceed with further simplification!
binomial factorization
Binomial factorization involves expressing a binomial (an algebraic expression with two terms) as a product of other simpler expressions. After grouping terms and factoring out the GCF, the expression becomes: \(15p(q - 1) + 12(q - 1)\).
Notice that \(q - 1\) is common in both terms. This is where binomial factorization comes into play.
We can factor \((q - 1)\) out from both terms, so the expression becomes: \((q - 1)(15p + 12)\).
In essence, binomial factorization helps break down complex expressions into simpler, more manageable parts.
This final step makes it very clear that \((q - 1)\) is multiplied by \(15p + 12\)!
Notice that \(q - 1\) is common in both terms. This is where binomial factorization comes into play.
We can factor \((q - 1)\) out from both terms, so the expression becomes: \((q - 1)(15p + 12)\).
In essence, binomial factorization helps break down complex expressions into simpler, more manageable parts.
This final step makes it very clear that \((q - 1)\) is multiplied by \(15p + 12\)!
elementary algebra
Elementary algebra is the foundation of algebra that deals with simple operations and the basic solving methods.
It involves the use of numbers, variables, and basic arithmetic operations like addition, subtraction, multiplication, and division.
In the given exercise, we use elementary algebra to factor the expression completely.
Starting with the expression \(15pq - 15p + 12q - 12\):
- Group the terms: \((15pq - 15p) + (12q - 12)\).
- Factor out the greatest common factor in each group: \(15p(q - 1) + 12(q - 1)\).
- Factor out the common binomial factor: \((q - 1)(15p + 12)\).
This process incorporates the basic operations and principles of elementary algebra, making it approachable and understandable!
It involves the use of numbers, variables, and basic arithmetic operations like addition, subtraction, multiplication, and division.
In the given exercise, we use elementary algebra to factor the expression completely.
Starting with the expression \(15pq - 15p + 12q - 12\):
- Group the terms: \((15pq - 15p) + (12q - 12)\).
- Factor out the greatest common factor in each group: \(15p(q - 1) + 12(q - 1)\).
- Factor out the common binomial factor: \((q - 1)(15p + 12)\).
This process incorporates the basic operations and principles of elementary algebra, making it approachable and understandable!