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Divide. Enter your answer as a completely simplified rational expression with nonnegative exponents.

[tex]\[ \frac{\left(\frac{3x + 8}{x^2 + 2x - 15}\right)}{\left(\frac{3x^2 - x - 24}{2x^2 + 15x + 25}\right)} \][/tex]

Sagot :

GinnyAnswer
To divide the given rational expressions, follow these steps:

1. Rewrite the Division as Multiplication:
When dividing two rational expressions, you can multiply by the reciprocal of the second expression:
[tex]\[ \frac{\left(\frac{3 x + 8}{x^2 + 2 x - 15}\right)}{\left(\frac{3 x^2 - x - 24}{2 x^2 + 15 x + 25}\right)} = \frac{3 x + 8}{x^2 + 2 x - 15} \times \frac{2 x^2 + 15 x + 25}{3 x^2 - x - 24} \][/tex]

2. Factor the Denominators if Possible:
[tex]\[ x^2 + 2 x - 15 = (x + 5)(x - 3) \][/tex]
[tex]\[ 2 x^2 + 15 x + 25 = (2 x + 5)(x + 5) \][/tex]

3. Replace Factored Expressions:
Substitute the factored forms into the expression:
[tex]\[ \frac{3 x + 8}{(x + 5)(x - 3)} \times \frac{(2 x + 5)(x + 5)}{3 x^2 - x - 24} \][/tex]

4. Factor Numerator and Denominator if Possible:
Let’s assume [tex]\(3 x^2 - x - 24\)[/tex] can be factored:
[tex]\[ 3 x^2 - x - 24 = (3 x + 8)(x - 3) \][/tex]

5. Substitute back into the expression:
[tex]\[ \frac{3 x + 8}{(x + 5)(x - 3)} \times \frac{(2 x + 5)(x + 5)}{(3 x + 8)(x - 3)} \][/tex]

6. Cancel Common Factors:
Cancel out the common factors [tex]\((3 x + 8)\)[/tex], [tex]\((x + 5)\)[/tex], and [tex]\((x - 3)\)[/tex] from the numerator and the denominator:
[tex]\[ \frac{(2 x + 5)}{(x - 3)} \][/tex]

8. Simplified Result:
The final result of the division is:
[tex]\[ \frac{2 x + 5}{x^2 - 6x + 9} \][/tex]

Therefore, the completely simplified rational expression is:
[tex]\[ \boxed{\frac{2 x + 5}{x^2 - 6 x + 9}} \][/tex]
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