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Sagot :
To solve the problem of finding the product of the rational expressions [tex]\(\frac{2}{x+1} \cdot \frac{5}{3x}\)[/tex], we'll follow several steps:
1. Numerator Multiplication:
Multiply the numerators of the given rational expressions:
[tex]\[ \text{Numerator} = 2 \cdot 5 = 10 \][/tex]
2. Denominator Multiplication:
Multiply the denominators of the given rational expressions:
[tex]\[ \text{Denominator} = (x + 1) \cdot (3x) \][/tex]
3. Expand the Denominator:
Distribute [tex]\(3x\)[/tex] into [tex]\((x + 1)\)[/tex]:
[tex]\[ (x + 1) \cdot 3x = 3x \cdot x + 3x \cdot 1 = 3x^2 + 3x \][/tex]
4. Form the Resulting Rational Expression:
Combine the results for the numerator and the denominator:
[tex]\[ \frac{10}{3x^2 + 3x} \][/tex]
Thus, the product of [tex]\(\frac{2}{x+1} \cdot \frac{5}{3x}\)[/tex] is [tex]\(\frac{10}{3x^2 + 3x}\)[/tex].
Therefore, the correct answer is:
C. [tex]\(\frac{10}{3x^2 + 3x}\)[/tex]
1. Numerator Multiplication:
Multiply the numerators of the given rational expressions:
[tex]\[ \text{Numerator} = 2 \cdot 5 = 10 \][/tex]
2. Denominator Multiplication:
Multiply the denominators of the given rational expressions:
[tex]\[ \text{Denominator} = (x + 1) \cdot (3x) \][/tex]
3. Expand the Denominator:
Distribute [tex]\(3x\)[/tex] into [tex]\((x + 1)\)[/tex]:
[tex]\[ (x + 1) \cdot 3x = 3x \cdot x + 3x \cdot 1 = 3x^2 + 3x \][/tex]
4. Form the Resulting Rational Expression:
Combine the results for the numerator and the denominator:
[tex]\[ \frac{10}{3x^2 + 3x} \][/tex]
Thus, the product of [tex]\(\frac{2}{x+1} \cdot \frac{5}{3x}\)[/tex] is [tex]\(\frac{10}{3x^2 + 3x}\)[/tex].
Therefore, the correct answer is:
C. [tex]\(\frac{10}{3x^2 + 3x}\)[/tex]
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