Simplifying this fraction

**sstr** · June 9th 2006, 12:21 AM

My apologies.

64a^3 + b^3
___________
16 a^2 - b^2

Divided by

16a^2b^2 - 4ab^3 + b^4
_______________________
4a^2 - ab + 12a - 3b

Sorry I don't know how to use the math function yet.

**Soroban** · June 9th 2006, 02:17 AM

Hello, sstr!

A fascinating problem: four types of factoring are required.

$\frac{64a^3 + b^3}{16a^2 - b^2} \div \frac{16a^2b^2 - 4ab^3 + b^4}{4a^2 - ab + 12a - 3b}$

Sum of Cubes: . $64a^3 + b^3 \;= \;(4a + b)(16a^2 - 4ab + b^2)$

Difference of Squares: . $16a^2 - b^2 \;= \;(4a - b)(4a + b)$

Common factors: . $16a^2b^2 - 4ab^3 + b^4 \;= \;b^2(16a^2 - 4ab + b^2)$

Grouping: . $4a^2 - ab + 12a - 3b \;= \;a(4a - b) + 3(4a - b)\;=$ $\;(4a-b)(a+3)$

The problem becomes: . $\frac{(4a + b)(16a^2 - 4ab + b^2)}{(4a - b)(4a + b)}\; \times$ $\frac{(4a - b)(a + 3)}{b^2(16a^2 - 4ab + b^2)}$

. . . which reduces to: . $\frac{a + 3}{b^2}$

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