5xy^2 + 15x^2y^2 - 25xy^2 + 65x^2y^2 = 10x^2y^2 + 45xy^2
Factor out the GcF of each polynomial in the equation
To factor out the Greatest Common Factor (GCF) of each polynomial in the equation, we need to determine the factors that all terms have in common.
Let's start by looking at the given equation:
5xy^2 + 15x^2y^2 - 25xy^2 + 65x^2y^2 = 10x^2y^2 + 45xy^2
Step 1: Identify the terms
The equation has multiple terms separated by either addition or subtraction.
5xy^2, 15x^2y^2, -25xy^2, 65x^2y^2, 10x^2y^2, and 45xy^2 are the terms in this equation.
Step 2: Find the factors of each term
Now, we need to find the factors of the coefficients (numbers in front of the variables) and the variables in each term.
The factors of each term are as follows:
5xy^2 has a factor of 5.
15x^2y^2 has factors of 15, x^2, and y^2.
-25xy^2 has a factor of -25.
65x^2y^2 has factors of 65, x^2, and y^2.
10x^2y^2 has factors of 10, x^2, and y^2.
45xy^2 has factors of 45 and y^2.
Step 3: Determine the GCF
To identify the GCF, we need to find the highest common factor (highest number or variable that appears in all terms with its highest exponent).
The GCF based on the factors of each term is as follows:
5xy^2: The GCF here is 1 (no number or variable appears in all terms with a higher exponent).
15x^2y^2: The GCF here is x^2y^2.
-25xy^2: The GCF here is 1 (no number or variable appears in all terms with a higher exponent).
65x^2y^2: The GCF here is x^2y^2.
10x^2y^2: The GCF here is x^2y^2.
45xy^2: The GCF here is y^2.
Step 4: Factor out the GCF
Now that we have determined the GCF, we can factor it out from each term:
The factored equation becomes:
GCF * (5xy^2 + 15x^2y^2 - 25xy^2 + 65x^2y^2) = GCF * (10x^2y^2 + 45xy^2)
Replacing the GCF values:
x^2y^2 * (5xy^2 + 15x^2y^2 - 25xy^2 + 65x^2y^2) = x^2y^2 * (10x^2y^2 + 45xy^2)
Thus, the factored form is:
5xy^2 + 15x^2y^2 - 25xy^2 + 65x^2y^2 = 10x^2y^2 + 45xy^2