Question

1. 8y^2-4y+1 divided by 2y-1

2. 2a^2+a+3 divided by a-1
3. 6x^3+11x^2-4x-4 divided by 3x-2
4. 6x^2+11x-35 divided by 3x-5
5. x^2-4 divided by x-1
6. y^3-4y^2+7y-6 divided by y-2
7. x^3+y^3 divided by x-y
8. a^4+4b^4 divided by a^2-2ab+2b^2
9. a^2^n-a^n-6 divided by a^n+8

Answer 1

Q1. The answer is
`4y+ (1)/(2y-1)`


`(8y^(2)-4y+1 )/(2y-1) = (4y*2y-4y+1)/(2y-1) = (4y(2y-1)+1)/(2y-1) = (4y(2y-1))/(2y-1)+ (1)/(2y-1) =4y+ (1)/(2y-1)`


Q2. The answer is
`2a+3+ (6)/(a-1)`


`(2 a^(2)+a+3 )/(a-1) = (a*2a-2a+3a+3)/(a-1) = (2a(a-1)+3a+3)/(a-1)= (2a(a-1))/(a-1)+ (3a+3)/(a-1) = \\ \\ =2a+ (3a+3)/(a-1)=2a+ (3a-3+3+3)/(a-1)=2a+ (3(a-1)+6)/(a-1) =2a+ (3(a-1))/(a-1) + (6)/(a-1)= \\ \\ =2a+3+ (6)/(a-1)`


Q3. The answer is
`2 x^(2) +5x+2`


`(6 x^(3) +11 x^(2) -4x-4)/(3x-2) = (3x*2 x^(2)-2 x^(2) *2+15 x^(2) -4x-4 )/(3x-2) = (2 x^(2) (3x-2)+15 x^(2) -4x-4)/(3x-2)= \\ \\ = (2 x^(2) (3x-2))/(3x-2) + (15 x^(2) -4x-4)/(3x-2) =2 x^(2) +(15 x^(2) -4x-4)/(3x-2)=2 x^(2) + (15 x^(2) -10x+6x-4)/(3x-2)= \\ \\ =2 x^(2) + (5x*3x-5x*2+6x-4)/(3x-2) =2 x^(2) + (5x(3x-2)+3x*2-2*2)/(3x-2) = \\ \\ =2 x^(2) + (5x(3x-2))/(3x-2) + (3x*2-2*2)/(3x-2) =2 x^(2) +5x+ (2(3x-2))/(3x-2) =2 x^(2) +5x+2`


Q4. The answer is 2x + 7


`(6 x^(2) +11x-35)/(3x-5) = (6 x^(2) -10x+21x-35 )/(3x-5) = (3 x *2x-5*2x+7*3x-7*5 )/(3x-5) = \\ \\ = (2x(3x-5)+7(3x-5))/(3x-5)= = ((3x-5)(2x+7))/(3x-5) =2x+7`


Q5. The answer is
`x+1- (3)/(x-1)`


`( x^(2) -4)/(x-1) = ( x^(2) -x+x-1-3 )/(x-1) = (x*x-x+x-1-3)/(x-1) = (x(x-1)+(x-1)-3)/(x-1) = \\ \\ ((x+1)(x-1)-3)/(x-1) = ((x+1)(x-1))/(x-1) -(3)/(x-1) =x+1- (3)/(x-1)`


Q6. The answer is
`y^(2) -2y+3`


`( y^(3)-4 y^(2)+7y-6 )/(y-2) = (y* y^(2) -2y^(2)-2 y^(2) +7y-6 )/(y-2) = (y^(2)(y-2)-2 y^(2) +7y-6)/(y-2)= \\ \\ = (y^(2)(y-2))/(y-2)+ (-2 y^(2) +7y-6)/(y-2) = y^(2) + (-2 y^(2) +4y + 3y-6)/(y-2) = \\ \\ =y^(2) + (-2y*y-2y(-2)+3y-3*2)/(y-2) = y^(2) + ((-2y)(y-2)+3(y-2))/(y-2) = \\ \\ = y^(2) + ((-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}`


`( x^(3) - (x)/(y) y^(3) )/(x-y) = ((x-y)( x^(2) +xy+ y^(2)) )/(x-y) = ( x^(2) +xy+ y^(2))/(x-y)`


Q8. The answer is
`(a^(2) +2ab+2b^(2))`


`(a^(4) +4b^(4) )/(a^(2)-2ab+2 b^(2) ) = ( (a^(2))^(2) +(2b)^(2))/(a^(2)-2ab+2 b^(2)) = ((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}


`( (a^(2)) ^(n) - a^(n-6) )/( a^(n+8) )= ((a^(2)) ^(n) )/(a^(n+8))- ( a^(n-6) )/(a^(n+8)) \\ \\ (x^(y)) ^(z)= x^(y*z) \\ \\ ( x^(y) )/( x^(z) ) = x^(y-z) \\ \\ ((a^(2)) ^(n) )/(a^(n+8))- ( a^(n-6) )/(a^(n+8)) =(a^(2n) )/(a^(n+8))- ( 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)`