Answer :
Certainly! Let's break down the problem and go through it step by step.
### Part A: Factoring out the Greatest Common Factor (GCF)
To factor out the greatest common factor from the expression [tex]\(6x^2y - 3xy - 24xy^2 + 12y^2\)[/tex], we look for the largest expression that divides each term.
1. Identify the GCF of the coefficients: The coefficients are 6, -3, -24, and 12. The greatest common divisor of these numbers is 3.
2. Identify the GCF of the variables: Every term contains the variable [tex]\(y\)[/tex]. Therefore, [tex]\(y\)[/tex] is part of the GCF.
Thus, the GCF of the entire expression is [tex]\(3y\)[/tex].
3. Factor out the GCF:
[tex]\[ 6x^2y - 3xy - 24xy^2 + 12y^2 = 3y \left(\frac{6x^2y}{3y} - \frac{3xy}{3y} - \frac{24xy^2}{3y} + \frac{12y^2}{3y}\right) \][/tex]
Simplify the expression inside the parentheses:
[tex]\[ = 3y (2x^2 - x - 8xy + 4y) \][/tex]
Now we have factored out the greatest common factor [tex]\(3y\)[/tex].
So, the factored form in part A is:
[tex]\[ 3y(2x^2 - x - 8xy + 4y) \][/tex]
### Part B: Complete Factoring of the Entire Expression
Next, we'll factor the expression inside the parentheses completely, if possible.
1. Observe the quadratic form inside the parentheses:
[tex]\[ 2x^2 - x - 8xy + 4y \][/tex]
2. Group like terms together:
[tex]\[ = 2x^2 - 8xy - x + 4y \][/tex]
3. Factor by grouping:
Group the terms in pairs and factor each group:
[tex]\[ = 2x^2 - 8xy - x + 4y \\ = (2x^2 - 8xy) + (-x + 4y) \\ = 2x(x - 4y) - 1(x - 4y) \][/tex]
4. Factor out the common binomial factor [tex]\((x - 4y)\)[/tex]:
[tex]\[ = (2x - 1)(x - 4y) \][/tex]
5. Combine with the GCF from step A:
[tex]\[ = 3y(2x - 1)(x - 4y) \][/tex]
So, the completely factored form of the entire expression is:
[tex]\[ 3y(2x - 1)(x - 4y) \][/tex]
### Summary:
- Part A:
[tex]\[ 3y(2x^2 - x - 8xy + 4y) \][/tex]
- Part B:
[tex]\[ 3y(2x - 1)(x - 4y) \][/tex]
These steps provide a detailed breakdown of how to factor the expression completely.
### Part A: Factoring out the Greatest Common Factor (GCF)
To factor out the greatest common factor from the expression [tex]\(6x^2y - 3xy - 24xy^2 + 12y^2\)[/tex], we look for the largest expression that divides each term.
1. Identify the GCF of the coefficients: The coefficients are 6, -3, -24, and 12. The greatest common divisor of these numbers is 3.
2. Identify the GCF of the variables: Every term contains the variable [tex]\(y\)[/tex]. Therefore, [tex]\(y\)[/tex] is part of the GCF.
Thus, the GCF of the entire expression is [tex]\(3y\)[/tex].
3. Factor out the GCF:
[tex]\[ 6x^2y - 3xy - 24xy^2 + 12y^2 = 3y \left(\frac{6x^2y}{3y} - \frac{3xy}{3y} - \frac{24xy^2}{3y} + \frac{12y^2}{3y}\right) \][/tex]
Simplify the expression inside the parentheses:
[tex]\[ = 3y (2x^2 - x - 8xy + 4y) \][/tex]
Now we have factored out the greatest common factor [tex]\(3y\)[/tex].
So, the factored form in part A is:
[tex]\[ 3y(2x^2 - x - 8xy + 4y) \][/tex]
### Part B: Complete Factoring of the Entire Expression
Next, we'll factor the expression inside the parentheses completely, if possible.
1. Observe the quadratic form inside the parentheses:
[tex]\[ 2x^2 - x - 8xy + 4y \][/tex]
2. Group like terms together:
[tex]\[ = 2x^2 - 8xy - x + 4y \][/tex]
3. Factor by grouping:
Group the terms in pairs and factor each group:
[tex]\[ = 2x^2 - 8xy - x + 4y \\ = (2x^2 - 8xy) + (-x + 4y) \\ = 2x(x - 4y) - 1(x - 4y) \][/tex]
4. Factor out the common binomial factor [tex]\((x - 4y)\)[/tex]:
[tex]\[ = (2x - 1)(x - 4y) \][/tex]
5. Combine with the GCF from step A:
[tex]\[ = 3y(2x - 1)(x - 4y) \][/tex]
So, the completely factored form of the entire expression is:
[tex]\[ 3y(2x - 1)(x - 4y) \][/tex]
### Summary:
- Part A:
[tex]\[ 3y(2x^2 - x - 8xy + 4y) \][/tex]
- Part B:
[tex]\[ 3y(2x - 1)(x - 4y) \][/tex]
These steps provide a detailed breakdown of how to factor the expression completely.
