Sure, let's factor the given expression step-by-step by applying the distributive property and other factoring techniques.
We start with the expression:
[tex]\[ 3 x^2 y^2 (4 y - 5 x) \][/tex]
Here's the step-by-step process to factor this expression:
1. Identify the Common Factors:
The given expression is already in factored form. Let's break down the expression inside the parentheses first:
[tex]\[ 4 y - 5 x \][/tex]
This part doesn't appear to have any common factors besides 1, so it stays as is.
2. Multiply by the Monomial:
The expression also includes a monomial term, [tex]\( 3 x^2 y^2 \)[/tex]. To factor it properly, we need to remember that every term in the expression will be multiplied by this monomial:
[tex]\[ 3 x^2 y^2 \cdot (4 y - 5 x) \][/tex]
3. Distribute the Monomial:
To get the expression back without the parentheses, we distribute [tex]\( 3 x^2 y^2 \)[/tex]:
[tex]\[ 3 x^2 y^2 \cdot 4 y - 3 x^2 y^2 \cdot 5 x \][/tex]
4. Multiply Each Term:
Compute the multiplication for each term individually:
- For the first term: [tex]\( 3 x^2 y^2 \cdot 4 y \)[/tex]:
[tex]\[ 3 \cdot 4 \cdot x^2 \cdot y^2 \cdot y = 12 x^2 y^3 \][/tex]
- For the second term: [tex]\( 3 x^2 y^2 \cdot 5 x \)[/tex]:
[tex]\[ 3 \cdot (-5) \cdot x^2 \cdot y^2 \cdot x = -15 x^3 y^2 \][/tex]
5. Combine Terms:
Now we combine the two resulting terms to write the expanded expression:
[tex]\[ 12 x^2 y^3 - 15 x^3 y^2 \][/tex]
This is the expanded form of the given factored expression.
6. Re-Factoring:
To ensure we can go back to the original factored form, we take out the greatest common factor from the expanded expression, which we identified initially:
[tex]\[ 12 x^2 y^3 - 15 x^3 y^2 = 3 x^2 y^2 (4 y - 5 x) \][/tex]
Conclusively, after expanding and refactoring, we see that:
[tex]\[ 3 x^2 y^2 (4 y - 5 x) \][/tex]
is indeed the factored form of the given expression. The alternative way to arrange the factors while preserving the sign is:
[tex]\[ 3 x^2 y^2 (-5 x + 4 y) \][/tex]
Specifically:
[tex]\[ 3 x^2 y^2 (4 y - 5 x) = -3 x^2 y^2 (5 x - 4 y) \][/tex]
So, this expression has been factored correctly.
