Help with Subtracting Rational Expressions?

First Problem: BTW...... these are really long problems!!! So HERE WE GO!!

(y / 2y +4) - (3 / y +2) Given

(y^2 +2y) - (6y +12) / (2y +4)(y+2) Cross Multiply to obtain common denominator.

y^2 +2y -6y -12 / 2 (y +2) (y +2) Numerator: Distribute the negative sign

Denominator factor out a 2. Notice I left (y+2) as two seperate factors. It is so you can cancel them out in later steps.

y^2 -4y -12 / 2 (y +2) (y +2) Combine like terms.

(y -6)(y+2) / 2 (y +2) Factor out the numerator by X factoring.

y -6 / 2(y +2) Cancel out (y+2) from the numerator & the denominator. Can't simplify any further. Therefore, this is your answer!

Second Problem:

(x / 3x +9) - (8 / x^2 +3x) Given

(x^3 +3x^2) - (24x +9) / (3x +9) (x^2 +3x) Cross Multiply to obtain common denominator.

x^3 +3x^2 -24x -9 / (3x +9)(x^2 +3x) Distribute the negative sign.

x^2(x +3) -3(8x +3) / 3(x +3) x(x +3) Numerator: Factor our x^2 from (x^3 +3x^2) & 3 from (-24x -9) Denominator: Factor out 3 from (3x+9) & x from (x+3).

(x^2 -3)(x+3)(8x+3) / (x+3)(x+3)(x+3) Grouped X^2 & 3 in the numerator & groupeed x & 3 in the denominator

(x^2 -3)(8x+3) / (x+3)^2 Canceled out (x+3) from the numerator & denominator. Simplify the denominator & VOILA!!

Third Problem:

(3y / y^2 -25) - (8 / y-5) Given

(3y^2 -15y) - (8y^2 -200) / (y^2 -25)(y-5) Cross Multiply to obtain common denominator.

3y^2 -15y -8y^2 +200 / (y^2 -25)(y-5) Distribute negative sign in the numerator.

-5y^2 -15y +200 / (y^2 -25)(y-5) Combine like terms in the numerator.

-5(y^2 +3y -40) / (y+5)(y-5)(y-5) Factor out a -5 in the numerator. I split up (y^2 -25) in the denominator by using this property (a^2 + b^2) = (a+b)(a-b)

-5(y+8)(y-5) / (y+5)(y-5)(y-5) X factor the (y^2 +3y -40) in the numerator.

-5(y+8) / (y+5)(y-5) Canceled out (y-5) in the numerator & denominator. This is your final answer.

After these three problems you should be able to do the last one...