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Find the G.C.F of [tex][tex]$15\left(x^2 + 2xy + y^2\right), 10(x + y)^3,$[/tex][/tex] and [tex][tex]$25\left(x^2 - y^2\right)$[/tex][/tex].

A. [tex][tex]$10(x + y)$[/tex][/tex]
B. [tex][tex]$5(x + y)$[/tex][/tex]
C. [tex][tex]$5(x - y)$[/tex][/tex]
D. [tex][tex]$5(x + y)(x - y)$[/tex][/tex]

Sagot :

GinnyAnswer
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]
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