To find the greatest common factor (GCF) of the numerator and the denominator of the rational expression [tex]\(\frac{5x - 20}{x^2 - 2x - 8}\)[/tex], we first need to factor both the numerator and the denominator.
Step 1: Factor the numerator [tex]\(5x - 20\)[/tex]
We can factor out the common factor from both terms in the numerator:
[tex]\[ 5x - 20 = 5(x - 4) \][/tex]
Step 2: Factor the denominator [tex]\(x^2 - 2x - 8\)[/tex]
To factor the quadratic expression [tex]\(x^2 - 2x - 8\)[/tex], we look for two numbers that multiply to [tex]\(-8\)[/tex] (the constant term) and add up to [tex]\(-2\)[/tex] (the coefficient of the linear term). These numbers are [tex]\( -4 \)[/tex] and [tex]\( 2 \)[/tex].
So, we can write the quadratic as:
[tex]\[ x^2 - 2x - 8 = (x - 4)(x + 2) \][/tex]
Step 3: Identify the GCF
Now that we have factored both the numerator and the denominator, let's write them in their factored forms:
Numerator: [tex]\( 5(x - 4) \)[/tex]
Denominator: [tex]\( (x - 4)(x + 2) \)[/tex]
The common factor between the numerator and the denominator is [tex]\( x - 4 \)[/tex].
Therefore, the greatest common factor (GCF) of the numerator and the denominator is:
[tex]\[ x - 4 \][/tex]
So, the correct answer is:
A. [tex]\( x - 4 \)[/tex]
