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What is the difference of the rational expressions below?

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

A. [tex]\(\frac{6x-5}{-3}\)[/tex]
B. [tex]\(\frac{6x^2-5x-15}{-3}\)[/tex]
C. [tex]\(\frac{6x-5}{x^2-3x}\)[/tex]
D. [tex]\(\frac{6x^2-5x+15}{x^2-3x}\)[/tex]


Sagot :

To find the difference of the rational expressions [tex]\(\frac{6x}{x-3} - \frac{5}{x}\)[/tex], we will follow a step-by-step approach.

1. Identify the common denominator:
The denominators of the two fractions are [tex]\(x - 3\)[/tex] and [tex]\(x\)[/tex]. To subtract these fractions, we need a common denominator, which is the product of these denominators:
[tex]\[ (x - 3) \cdot x = x(x - 3) = x^2 - 3x \][/tex]

2. Rewrite each fraction with the common denominator:
[tex]\[ \frac{6x}{x - 3} = \frac{6x \cdot x}{(x - 3) \cdot x} = \frac{6x^2}{x^2 - 3x} \][/tex]
[tex]\[ \frac{5}{x} = \frac{5 \cdot (x - 3)}{x \cdot (x - 3)} = \frac{5(x - 3)}{x^2 - 3x} \][/tex]

3. Expand the second term's numerator:
[tex]\[ \frac{5(x - 3)}{x^2 - 3x} = \frac{5x - 15}{x^2 - 3x} \][/tex]

4. Form the equations with the common denominator:
[tex]\[ \frac{6x^2}{x^2 - 3x} - \frac{5x - 15}{x^2 - 3x} \][/tex]

5. Combine the numerators under the common denominator:
[tex]\[ \frac{6x^2 - (5x - 15)}{x^2 - 3x} \][/tex]

6. Simplify the numerator:
[tex]\[ 6x^2 - (5x - 15) = 6x^2 - 5x + 15 \][/tex]

7. Write the simplified form of the fraction:
[tex]\[ \frac{6x^2 - 5x + 15}{x^2 - 3x} \][/tex]

Thus, the difference of the rational expressions [tex]\(\frac{6x}{x-3} - \frac{5}{x}\)[/tex] is:

[tex]\[ \boxed{\frac{6x^2 - 5x + 15}{x^2 - 3x}} \][/tex]

The correct answer from the given options is:
D. [tex]\(\frac{6x^2 - 5x + 15}{x^2 - 3x}\)[/tex]