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To find the Greatest Common Factor (G.C.F) of the given expressions [tex]\(15(x^2 + 2xy + y^2)\)[/tex], [tex]\(10(x + y)^3\)[/tex], and [tex]\(25(x^2 - y^2)\)[/tex], let's proceed step-by-step.
### Step 1: Factor the given expressions
First, we will factor the individual expressions as fully as possible.
Expression 1:
[tex]\[ 15(x^2 + 2xy + y^2) \][/tex]
Notice that [tex]\(x^2 + 2xy + y^2\)[/tex] is a perfect square trinomial that can be factored as:
[tex]\[ x^2 + 2xy + y^2 = (x + y)^2 \][/tex]
Thus, the factored form of the first expression is:
[tex]\[ 15(x^2 + 2xy + y^2) = 15(x + y)^2 \][/tex]
Expression 2:
[tex]\[ 10(x + y)^3 \][/tex]
This expression is already given in its factored form.
Expression 3:
[tex]\[ 25(x^2 - y^2) \][/tex]
Notice that [tex]\(x^2 - y^2\)[/tex] is a difference of squares that can be factored as:
[tex]\[ x^2 - y^2 = (x + y)(x - y) \][/tex]
Thus, the factored form of the third expression is:
[tex]\[ 25(x^2 - y^2) = 25(x + y)(x - y) \][/tex]
### Step 2: Identify the common factors
Now, let’s identify the common factors among the factored expressions.
From the factorized forms:
1. [tex]\(15(x + y)^2\)[/tex]
2. [tex]\(10(x + y)^3\)[/tex]
3. [tex]\(25(x + y)(x - y)\)[/tex]
The common factor present in each term is [tex]\((x + y)\)[/tex].
### Step 3: Determine the G.C.F
Since [tex]\((x + y)\)[/tex] is a common factor, we need to find the highest power of [tex]\((x + y)\)[/tex] that is present in all terms.
- In [tex]\(15(x + y)^2\)[/tex], the term [tex]\((x + y)\)[/tex] appears to the power of 2.
- In [tex]\(10(x + y)^3\)[/tex], the term [tex]\((x + y)\)[/tex] appears to the power of 3.
- In [tex]\(25(x + y)(x - y)\)[/tex], the term [tex]\((x + y)\)[/tex] appears to the power of 1.
The smallest power of the common factor [tex]\((x + y)\)[/tex] is [tex]\( (x + y)^1 \)[/tex].
Additionally, note the coefficients: 15, 10, and 25. The greatest common divisor of these numbers is 5.
Therefore, combining the common factor and the GCD of the coefficients, the G.C.F of the given expressions is:
[tex]\[ 5(x + y) \][/tex]
Hence, the G.C.F is:
[tex]\[ \boxed{5(x + y)} \][/tex]
### Step 1: Factor the given expressions
First, we will factor the individual expressions as fully as possible.
Expression 1:
[tex]\[ 15(x^2 + 2xy + y^2) \][/tex]
Notice that [tex]\(x^2 + 2xy + y^2\)[/tex] is a perfect square trinomial that can be factored as:
[tex]\[ x^2 + 2xy + y^2 = (x + y)^2 \][/tex]
Thus, the factored form of the first expression is:
[tex]\[ 15(x^2 + 2xy + y^2) = 15(x + y)^2 \][/tex]
Expression 2:
[tex]\[ 10(x + y)^3 \][/tex]
This expression is already given in its factored form.
Expression 3:
[tex]\[ 25(x^2 - y^2) \][/tex]
Notice that [tex]\(x^2 - y^2\)[/tex] is a difference of squares that can be factored as:
[tex]\[ x^2 - y^2 = (x + y)(x - y) \][/tex]
Thus, the factored form of the third expression is:
[tex]\[ 25(x^2 - y^2) = 25(x + y)(x - y) \][/tex]
### Step 2: Identify the common factors
Now, let’s identify the common factors among the factored expressions.
From the factorized forms:
1. [tex]\(15(x + y)^2\)[/tex]
2. [tex]\(10(x + y)^3\)[/tex]
3. [tex]\(25(x + y)(x - y)\)[/tex]
The common factor present in each term is [tex]\((x + y)\)[/tex].
### Step 3: Determine the G.C.F
Since [tex]\((x + y)\)[/tex] is a common factor, we need to find the highest power of [tex]\((x + y)\)[/tex] that is present in all terms.
- In [tex]\(15(x + y)^2\)[/tex], the term [tex]\((x + y)\)[/tex] appears to the power of 2.
- In [tex]\(10(x + y)^3\)[/tex], the term [tex]\((x + y)\)[/tex] appears to the power of 3.
- In [tex]\(25(x + y)(x - y)\)[/tex], the term [tex]\((x + y)\)[/tex] appears to the power of 1.
The smallest power of the common factor [tex]\((x + y)\)[/tex] is [tex]\( (x + y)^1 \)[/tex].
Additionally, note the coefficients: 15, 10, and 25. The greatest common divisor of these numbers is 5.
Therefore, combining the common factor and the GCD of the coefficients, the G.C.F of the given expressions is:
[tex]\[ 5(x + y) \][/tex]
Hence, the G.C.F is:
[tex]\[ \boxed{5(x + y)} \][/tex]
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