Problem 47
Question
Factor by grouping. \(u v-9 u+2 v-18\)
Step-by-Step Solution
Verified Answer
(v - 9)(u + 2)
1Step 1 - Group Terms
Group the given terms into two pairs: (uv - 9u) and (2v - 18).
2Step 2 - Factor Out the GCD of Each Pair
Identify the greatest common divisor (GCD) in each pair and factor it out. For the first pair (uv - 9u), the GCD is u: u(v - 9). For the second pair (2v - 18), the GCD is 2: 2(v - 18/2) = 2(v - 9).
3Step 3 - Factor Out the Common Binomial
Now both groups have a common binomial factor, (v - 9): u(v - 9) + 2(v - 9). Factor out (v - 9): (v - 9)(u + 2).
Key Concepts
greatest common divisorbinomial factoralgebraic factoring
greatest common divisor
When factoring expressions, finding the greatest common divisor (GCD) is key. The GCD is the largest factor shared by all terms in a group.
For example, in the expression \(uv - 9u\), both terms contain the variable 'u'. Thus, 'u' is the GCD of this pair, making it possible to factor 'u' out:
\[ u(v - 9) \]
Similarly, for the expression \(2v - 18\), both terms share a numerical factor of 2. Thus, 2 is the GCD of this pair, allowing us to factor 2 out:
\[ 2(v - 9) \]
Identifying the GCD correctly simplifies your work and helps to find common factors easily.
Remember, the GCD must be a factor of all terms in the group, which helps in breaking down complex expressions in a simpler form.
For example, in the expression \(uv - 9u\), both terms contain the variable 'u'. Thus, 'u' is the GCD of this pair, making it possible to factor 'u' out:
\[ u(v - 9) \]
Similarly, for the expression \(2v - 18\), both terms share a numerical factor of 2. Thus, 2 is the GCD of this pair, allowing us to factor 2 out:
\[ 2(v - 9) \]
Identifying the GCD correctly simplifies your work and helps to find common factors easily.
Remember, the GCD must be a factor of all terms in the group, which helps in breaking down complex expressions in a simpler form.
binomial factor
A binomial factor refers to an expression involving a sum or difference of two terms that can be factored out of a larger expression.
Consider the factored pairs from the problem: \( u(v - 9) \) and \( 2(v - 9) \). Both expressions share the binomial factor \((v - 9)\).
This common binomial factor can be further factored out to show the expression in a completely factored form:
\[ u(v - 9) + 2(v - 9) \rightarrow (v - 9)(u + 2) \]
Binomial factors are convenient as they often appear when performing algebraic factoring, allowing multiple terms to be combined into simpler product forms.
Finding and factoring out common binomial factors simplifies expressions and reveals hidden commonalities between different terms.
Consider the factored pairs from the problem: \( u(v - 9) \) and \( 2(v - 9) \). Both expressions share the binomial factor \((v - 9)\).
This common binomial factor can be further factored out to show the expression in a completely factored form:
\[ u(v - 9) + 2(v - 9) \rightarrow (v - 9)(u + 2) \]
Binomial factors are convenient as they often appear when performing algebraic factoring, allowing multiple terms to be combined into simpler product forms.
Finding and factoring out common binomial factors simplifies expressions and reveals hidden commonalities between different terms.
algebraic factoring
Algebraic factoring involves breaking down algebraic expressions into their simplest components, or factors.
One effective method used in this problem is factoring by grouping.
First, expressions are grouped into pairs, like so:
(uv - 9u) and (2v - 18).
Next, each pair is factored by their GCD:
\[ u(v - 9) \text { and } 2(v - 9) \]
Finally, any common binomial factors are factored out. This process reveals the original expression as a product of simpler expressions:
\[ (v - 9)(u + 2) \]
This method is powerful because it leverages the shared factors between groups to simplify complex algebraic expressions.
Algebraic factoring not only simplifies expressions but also aids in solving equations and understanding mathematical relationships between terms.
One effective method used in this problem is factoring by grouping.
First, expressions are grouped into pairs, like so:
(uv - 9u) and (2v - 18).
Next, each pair is factored by their GCD:
\[ u(v - 9) \text { and } 2(v - 9) \]
Finally, any common binomial factors are factored out. This process reveals the original expression as a product of simpler expressions:
\[ (v - 9)(u + 2) \]
This method is powerful because it leverages the shared factors between groups to simplify complex algebraic expressions.
Algebraic factoring not only simplifies expressions but also aids in solving equations and understanding mathematical relationships between terms.