Verified answer

6.17
2) [tex]\frac{a^{4} -b^{4} }{ab^{2}-a^{3} }=\frac{(a^{2}-b^{2})(a^{2}+b^{2})}{a(b^{2}-a^{2})} =\frac{-(b^{2}-a^{2})(a^{2}+b^{2})}{a(b^{2}-a^{2})}=\frac{-(a^{2}+b^{2})}{a} = -\frac{a^{2}+b^{2} }{a}[/tex]
4) [tex]\frac{a^{4} - b^{4} }{a^{3} - b^{3}} =\frac{(a^{2} - b^{2})(a^{2} + b^{2})}{(a-b)(a^{2}+ab+b^{2})} =\frac{(a-b)(a+b)(a^{2} + b^{2})}{(a-b)(a^{2}+ab+b^{2})} = \frac{(a+b)(a^{2}+b^{2})}{a^{2}+ab+b^{2}} = \frac{a^{3}+ab^{2}+a^{2} b+b^{3}}{a^{2}+ab+b^{2}}[/tex]

6.18
2) [tex]\frac{1-2x+x^{2} }{x^{2}-1} =\frac{(1-x)^{2} }{(x-1)(x+1)}=\frac{(x-1)^{2} }{(x-1)(x+1)}=\frac{x-1}{x+1}[/tex]
4) [tex]\frac{a^{4}-b^{4}}{a^{2} -b^{2} } =\frac{(a^{2}-b^{2})(a^{2}+b^{2} )}{a^{2}-b^{2} } =a^{2} +b^{2}[/tex]

6.19
2) [tex]\frac{x^{6} -x^{8} }{x^{4} -x^{2}}=\frac{x^{2} (x^{4} -x^{6} )}{x^{2} (x^{2} -1)}=\frac{x^{4}(1-x^{2})}{x^{2} -1} =\frac{x^{4}(-(x^{2}-1))}{x^{2} -1}=-x^{4}[/tex]
4) [tex]\frac{a^{6}-a^{4}}{a^{3}+a^{2}}=\frac{a^2(a^4-a^2)}{a^2(a+1)} =\frac{a^2(a^2-1)}{a+1} = \frac{a^2(a-1)(a+1)}{a+1} =a^2(a-1)=a^3-a^2[/tex]