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(i)
(b^(2)-25)/(2b+10)÷((b-5)^(2))/(5b^(2))

(i) $\frac{b^{2}-25}{2 b+10} \div \frac{(b-5)^{2}}{5 b^{2}}$

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Divide polynomials using long division

Full solution

Q. (i)

\frac{b^{2}-25}{2 b+10} \div \frac{(b-5)^{2}}{5 b^{2}}

Recognize Division Rule: First, we need to recognize that dividing by a fraction is the same as multiplying by its reciprocal. So, we will rewrite the division as a multiplication by the reciprocal of the second fraction.
Rewrite as Multiplication: Rewrite the expression as a multiplication by the reciprocal of the second fraction: $\left(\frac{b^2 - 25}{2b + 10}\right) \times \left(\frac{5b^2}{(b - 5)^2}\right)$
Factor Numerator and Denominator: Next, we can factor the numerator of the first fraction and the denominator of the second fraction: $((b - 5)(b + 5) / (2b + 10)) \times (5b^2 / (b - 5)(b - 5))$
Cancel Common Factors: Now, we can simplify the expression by canceling out common factors. The $(b - 5)$ term in the numerator of the first fraction and one $(b - 5)$ term in the denominator of the second fraction cancel out. Also, $(2b + 10)$ in the denominator of the first fraction can be factored out as $2(b + 5)$ , which cancels with the $(b + 5)$ in the numerator.
Simplify Expression: After canceling the common factors, the expression simplifies to: $\frac{1}{2} \times \frac{5b^2}{b - 5}$
Multiply Remaining Terms: Now, multiply the remaining terms: $\frac{5}{2} \times \frac{b^2}{b - 5}$
Final Simplified Form: Finally, we can leave the expression in its simplified form: $\frac{5b^2}{2(b - 5)}$

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