Q1. The answer is \(4y+ \frac{1}{2y-1}\)
\(\frac{8y^{2}-4y+1 }{2y-1} = \frac{4y*2y-4y+1}{2y-1} = \frac{4y(2y-1)+1}{2y-1} = \frac{4y(2y-1)}{2y-1}+ \frac{1}{2y-1} =4y+ \frac{1}{2y-1}\)
Q2. The answer is \(2a+3+ \frac{6}{a-1}\)
\(\frac{2 a^{2}+a+3 }{a-1} = \frac{a*2a-2a+3a+3}{a-1} = \frac{2a(a-1)+3a+3}{a-1}= \frac{2a(a-1)}{a-1}+ \frac{3a+3}{a-1} = \\ \\ =2a+ \frac{3a+3}{a-1}=2a+ \frac{3a-3+3+3}{a-1}=2a+ \frac{3(a-1)+6}{a-1} =2a+ \frac{3(a-1)}{a-1} + \frac{6}{a-1}= \\ \\ =2a+3+ \frac{6}{a-1}\)
Q3. The answer is \(2 x^{2} +5x+2\)
\(\frac{6 x^{3} +11 x^{2} -4x-4}{3x-2} = \frac{3x*2 x^{2}-2 x^{2} *2+15 x^{2} -4x-4 }{3x-2} = \frac{2 x^{2} (3x-2)+15 x^{2} -4x-4}{3x-2}= \\ \\ = \frac{2 x^{2} (3x-2)}{3x-2} + \frac{15 x^{2} -4x-4}{3x-2} =2 x^{2} +\frac{15 x^{2} -4x-4}{3x-2}=2 x^{2} + \frac{15 x^{2} -10x+6x-4}{3x-2}= \\ \\ =2 x^{2} + \frac{5x*3x-5x*2+6x-4}{3x-2} =2 x^{2} + \frac{5x(3x-2)+3x*2-2*2}{3x-2} = \\ \\ =2 x^{2} + \frac{5x(3x-2)}{3x-2} + \frac{3x*2-2*2}{3x-2} =2 x^{2} +5x+ \frac{2(3x-2)}{3x-2} =2 x^{2} +5x+2\)
Q4. The answer is 2x + 7
\(\frac{6 x^{2} +11x-35}{3x-5} = \frac{6 x^{2} -10x+21x-35 }{3x-5} = \frac{3 x *2x-5*2x+7*3x-7*5 }{3x-5} = \\ \\ = \frac{2x(3x-5)+7(3x-5)}{3x-5}= = \frac{(3x-5)(2x+7)}{3x-5} =2x+7\)
Q5. The answer is \(x+1- \frac{3}{x-1}\)
\(\frac{ x^{2} -4}{x-1} = \frac{ x^{2} -x+x-1-3 }{x-1} = \frac{x*x-x+x-1-3}{x-1} = \frac{x(x-1)+(x-1)-3}{x-1} = \\ \\ \frac{(x+1)(x-1)-3}{x-1} = \frac{(x+1)(x-1)}{x-1} -\frac{3}{x-1} =x+1- \frac{3}{x-1}\)
Q6. The answer is \(y^{2} -2y+3\)
\(\frac{ y^{3}-4 y^{2}+7y-6 }{y-2} = \frac{y* y^{2} -2y^{2}-2 y^{2} +7y-6 }{y-2} = \frac{y^{2}(y-2)-2 y^{2} +7y-6}{y-2}= \\ \\ = \frac{y^{2}(y-2)}{y-2}+ \frac{-2 y^{2} +7y-6}{y-2} = y^{2} + \frac{-2 y^{2} +4y + 3y-6}{y-2} = \\ \\ =y^{2} + \frac{-2y*y-2y(-2)+3y-3*2}{y-2} = y^{2} + \frac{(-2y)(y-2)+3(y-2)}{y-2} = \\ \\ = y^{2} + \frac{(-2y+3)(y-2)}{y-2} = y^{2} +(-2y+3) =y^{2} -2y+3\)
Q7. The answer is \(x^{2} +xy+ y^{2}}{x-y}\)
\(\frac{ x^{3} - \frac{x}{y} y^{3} }{x-y} = \frac{(x-y)( x^{2} +xy+ y^{2}) }{x-y} = \frac{ x^{2} +xy+ y^{2}}{x-y}\)
Q8. The answer is \((a^{2} +2ab+2b^{2})\)
\(\frac{a^{4} +4b^{4} }{a^{2}-2ab+2 b^{2} } = \frac{ (a^{2})^{2} +(2b)^{2}}{a^{2}-2ab+2 b^{2}} = \frac{(a^{2} -2ab+2b^{2})(a^{2} +2ab+2b^{2}) }{(a^{2} -2ab+2b^{2})} =(a^{2} +2ab+2b^{2})\)
Q9. The answer is a^{n-8} - a^{-14}
\(\frac{ (a^{2}) ^{n} - a^{n-6} }{ a^{n+8} }= \frac{(a^{2}) ^{n} }{a^{n+8}}- \frac{ a^{n-6} }{a^{n+8}} \\ \\ (x^{y}) ^{z}= x^{y*z} \\ \\ \frac{ x^{y} }{ x^{z} } = x^{y-z} \\ \\
\frac{(a^{2}) ^{n} }{a^{n+8}}- \frac{ a^{n-6} }{a^{n+8}} =\frac{a^{2n} }{a^{n+8}}- \frac{ a^{n-6} }{a^{n+8}} = a^{2n-(n+8)} - a^{n-6-(n+8)} = \\ \\ =a^{2n-n-8} - a^{n-6-n-8} = a^{n-8} - a^{-14}\)
