What Is The Completely Factored Form Of $12xy - 9x - 8y + 6$?A. $(3x - 2)(4y - 3)$ B. $3x(4y - 3) - 2(4y + 3)$ C. $ − 6 X ( 4 Y − 3 ) ( 4 Y − 3 ) -6x(4y - 3)(4y - 3) − 6 X ( 4 Y − 3 ) ( 4 Y − 3 ) [/tex] D. $(3x - 2)(4y - 3)(4y - 3)$
Understanding the Problem
The given expression is a quadratic expression in two variables, x and y. The completely factored form of an expression is a product of its factors, where each factor is a linear or quadratic expression. To find the completely factored form of the given expression, we need to factorize it using the appropriate techniques.
Step 1: Factorize the Expression
The given expression is $12xy - 9x - 8y + 6$. We can start by factoring out the greatest common factor (GCF) of the terms. The GCF of the terms is 3, so we can factor out 3 from each term.
Factoring Out the GCF
Step 2: Factorize the Expression Inside the Parentheses
Now, we need to factorize the expression inside the parentheses. We can start by looking for two numbers whose product is 4 and whose sum is -3. These numbers are -4 and 1, so we can write:
Step 3: Factorize the Remaining Terms
Now, we need to factorize the remaining terms. We can start by looking for two numbers whose product is -8/3 and whose sum is -8/3. These numbers are -8/3 and 0, so we can write:
Step 4: Combine the Factors
Now, we can combine the factors we have obtained so far:
Step 5: Factorize the Common Factor
Now, we can factorize the common factor (4y - 3) from each term:
Step 6: Simplify the Expression
Now, we can simplify the expression by multiplying the factors:
Conclusion
The completely factored form of the given expression is $(3x - 2)(4y - 3)$. This is the correct answer.
Comparison with the Options
Now, let's compare our answer with the options:
- Option A: $(3x - 2)(4y - 3)$
- Option B: $3x(4y - 3) - 2(4y + 3)$
- Option C: $-6x(4y - 3)(4y - 3)$
- Option D: $(3x - 2)(4y - 3)(4y - 3)$
Our answer matches with option A.
Final Answer
The completely factored form of $12xy - 9x - 8y + 6$ is $(3x - 2)(4y - 3)$.
Understanding the Problem
The given expression is a quadratic expression in two variables, x and y. The completely factored form of an expression is a product of its factors, where each factor is a linear or quadratic expression. To find the completely factored form of the given expression, we need to factorize it using the appropriate techniques.
Step 1: Factorize the Expression
The given expression is $12xy - 9x - 8y + 6$. We can start by factoring out the greatest common factor (GCF) of the terms. The GCF of the terms is 3, so we can factor out 3 from each term.
Factoring Out the GCF
Step 2: Factorize the Expression Inside the Parentheses
Now, we need to factorize the expression inside the parentheses. We can start by looking for two numbers whose product is 4 and whose sum is -3. These numbers are -4 and 1, so we can write:
Step 3: Factorize the Remaining Terms
Now, we need to factorize the remaining terms. We can start by looking for two numbers whose product is -8/3 and whose sum is -8/3. These numbers are -8/3 and 0, so we can write:
Step 4: Combine the Factors
Now, we can combine the factors we have obtained so far:
Step 5: Factorize the Common Factor
Now, we can factorize the common factor (4y - 3) from each term:
Step 6: Simplify the Expression
Now, we can simplify the expression by multiplying the factors:
Conclusion
The completely factored form of the given expression is $(3x - 2)(4y - 3)$. This is the correct answer.
Comparison with the Options
Now, let's compare our answer with the options:
- Option A: $(3x - 2)(4y - 3)$
- Option B: $3x(4y - 3) - 2(4y + 3)$
- Option C: $-6x(4y - 3)(4y - 3)$
- Option D: $(3x - 2)(4y - 3)(4y - 3)$
Our answer matches with option A.
Final Answer
The completely factored form of $12xy - 9x - 8y + 6$ is $(3x - 2)(4y - 3)$.
Q&A: Completely Factored Form of $12xy - 9x - 8y + 6$
Q: What is the completely factored form of $12xy - 9x - 8y + 6$?
A: The completely factored form of $12xy - 9x - 8y + 6$ is $(3x - 2)(4y - 3)$.
Q: How do I factorize the expression $12xy - 9x - 8y + 6$?
A: To factorize the expression $12xy - 9x - 8y + 6$, we need to factor out the greatest common factor (GCF) of the terms. The GCF of the terms is 3, so we can factor out 3 from each term.
Q: What is the greatest common factor (GCF) of the terms in the expression $12xy - 9x - 8y + 6$?
A: The greatest common factor (GCF) of the terms in the expression $12xy - 9x - 8y + 6$ is 3.
Q: How do I factorize the expression inside the parentheses in the expression $3(4xy - 3x - \frac{8}{3}y + 2)$?
A: To factorize the expression inside the parentheses in the expression $3(4xy - 3x - \frac{8}{3}y + 2)$, we need to look for two numbers whose product is 4 and whose sum is -3. These numbers are -4 and 1, so we can write:
Q: How do I factorize the remaining terms in the expression $3(4xy - 3x - \frac{8}{3}y + 2)$?
A: To factorize the remaining terms in the expression $3(4xy - 3x - \frac{8}{3}y + 2)$, we need to look for two numbers whose product is -8/3 and whose sum is -8/3. These numbers are -8/3 and 0, so we can write:
Q: How do I combine the factors in the expression $3(x(4y - 3) - \frac{2}{3}(4y - 3))$?
A: To combine the factors in the expression $3(x(4y - 3) - \frac{2}{3}(4y - 3))$, we can factorize the common factor (4y - 3) from each term:
Q: How do I simplify the expression $3(4y - 3)(x - \frac{2}{3})$?
A: To simplify the expression $3(4y - 3)(x - \frac{2}{3})$, we can multiply the factors:
Q: What is the completely factored form of $12xy - 9x - 8y + 6$?
A: The completely factored form of $12xy - 9x - 8y + 6$ is $(3x - 2)(4y - 3)$.
Conclusion
The completely factored form of $12xy - 9x - 8y + 6$ is $(3x - 2)(4y - 3)$. This is the correct answer. We hope this article has helped you understand the completely factored form of the given expression. If you have any further questions or need help with any other math problems, feel free to ask!