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What is the greatest common factor (GCF) of the numerator and denominator in the rational expression below?

[tex]\[
\frac{9x + 45}{x^2 + 9x + 20}
\][/tex]

A. 20
B. 9
C. [tex]\(x + 4\)[/tex]
D. [tex]\(x + 5\)[/tex]

Sagot :

GinnyAnsweridn
To find the greatest common factor (GCF) of the given rational expression:

[tex]\[ \frac{9x + 45}{x^2 + 9x + 20} \][/tex]

we need to first factor both the numerator and the denominator.

### Step 1: Factor the Numerator

The numerator is [tex]\(9x + 45\)[/tex].

Notice that both terms have a common factor:

[tex]\[ 9x + 45 = 9(x + 5) \][/tex]

### Step 2: Factor the Denominator

The denominator is [tex]\(x^2 + 9x + 20\)[/tex].

To factor this quadratic expression, we look for two numbers that multiply to 20 (the constant term) and add up to 9 (the coefficient of [tex]\(x\)[/tex]).

The numbers that fulfill these requirements are 4 and 5. Therefore, we can factor the quadratic as:

[tex]\[ x^2 + 9x + 20 = (x + 4)(x + 5) \][/tex]

### Step 3: Identify the Common Factor

Now that we have factored both the numerator and the denominator, we can see the expressions in their factored forms:

The numerator is [tex]\(9(x + 5)\)[/tex] and the denominator is [tex]\((x + 4)(x + 5)\)[/tex].

Both the numerator and the denominator share a common factor of [tex]\(x + 5\)[/tex].

### Step 4: Conclusion

Hence, the greatest common factor (GCF) of the numerator and the denominator is:

[tex]\[ \boxed{x + 5} \][/tex]
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