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What is the simplified form of the rational expression below?

[tex]\[ \frac{6x^2 - 54}{5x^2 + 15x} \][/tex]

A. [tex]\(\frac{6(x-3)}{5x}\)[/tex]

B. [tex]\(\frac{6(x+3)}{5}\)[/tex]

C. [tex]\(\frac{6(x-3)}{5}\)[/tex]

D. [tex]\(\frac{6(x+3)}{5x}\)[/tex]


Sagot :

To simplify the rational expression [tex]\(\frac{6x^2 - 54}{5x^2 + 15x}\)[/tex], we need to factor both the numerator and the denominator, and then simplify by cancelling common factors.

1. Factor the Numerator:
The numerator is [tex]\(6x^2 - 54\)[/tex].

Notice that [tex]\(6x^2 - 54\)[/tex] has a common factor of 6. So, we can factor out the 6:
[tex]\[ 6x^2 - 54 = 6(x^2 - 9) \][/tex]

Notice that [tex]\(x^2 - 9\)[/tex] is a difference of squares, which can be factored further:
[tex]\[ x^2 - 9 = (x - 3)(x + 3) \][/tex]

So, the numerator fully factored is:
[tex]\[ 6(x - 3)(x + 3) \][/tex]

2. Factor the Denominator:
The denominator is [tex]\(5x^2 + 15x\)[/tex].

Notice that [tex]\(5x^2 + 15x\)[/tex] has a common factor of [tex]\(5x\)[/tex]. So, we can factor out the [tex]\(5x\)[/tex]:
[tex]\[ 5x^2 + 15x = 5x(x + 3) \][/tex]

3. Write the Rational Expression with the Factors:
Now that we have factored both the numerator and the denominator, we can write:
[tex]\[ \frac{6x^2 - 54}{5x^2 + 15x} = \frac{6(x - 3)(x + 3)}{5x(x + 3)} \][/tex]

4. Simplify by Cancelling Common Factors:
We notice that [tex]\((x + 3)\)[/tex] is a common factor in both the numerator and the denominator, so we can cancel [tex]\((x + 3)\)[/tex] from both:
[tex]\[ \frac{6(x - 3)\cancel{(x + 3)}}{5x\cancel{(x + 3)}} = \frac{6(x - 3)}{5x} \][/tex]

So, the simplified form of the given rational expression is:

[tex]\[ \frac{6(x - 3)}{5x} \][/tex]

Therefore, the correct answer is:

A. [tex]\(\frac{6(x - 3)}{5x}\)[/tex]