91Ó°ÊÓ

After finding the simplest form of each fraction, explain why it cannot be simplified more. (a) \(\frac{2 x+3}{2 x+6}\) (b) \(\frac{2(x+6)}{2 x+6}\)

Evaluate the given expressions by using factoring. The results may be checked with a calculator. $$\frac{5^{9}-5^{7}}{7^{2}-5^{2}}$$

Find constants \(A\) and \(B\) such that the equation is true. $$\frac{x-12}{x^{2}+x-6}=\frac{A}{x+3}+\frac{B}{x-2}$$

Perform the indicated operations. Each expression occurs in the indicated area of application. $$\left(\frac{3 P x}{2 L^{2}}\right)^{2}+\left(\frac{P}{2 L}\right)^{2} \text { (force of a weld) }$$

Give the required explanations. Factor \(n^{2}+n,\) and then explain why it represents a positive even integer if \(n\) is a positive integer.

Factor the given expressions completely. Each is from the technical area indicated. $$3 A d u^{2}-4 A d u v+A d v^{2} \quad \text { (water power) }$$

After finding the simplest form of each fraction, explain why it cannot be simplified more. (a) \(\frac{x^{2}-x-2}{x^{2}-x}\) (b) \(\frac{x^{2}-x-2}{x^{2}+x}\)

Factor the given expressions completely. Each is from the technical area indicated. $$k^{2} A^{2}+2 k \lambda A+\lambda^{2}-\alpha^{2} \quad \text { (robotics) }$$

Find constants \(A\) and \(B\) such that the equation is true. $$\frac{23-x}{2 x^{2}+7 x-4}=\frac{A}{2 x-1}-\frac{B}{x+4}$$

Give the required explanations. Factor \(n^{3}-n,\) and then explain why it represents a multiple of 6 if \(n\) is an integer greater than 1