Question 149627
# 1


{{{1/(x-2) - 4/(x^2) + 3/(x+2)}}} Start with the given expression.



{{{((x^2(x+2))/(x^2(x+2)))(1/(x-2)) - (((x+2)(x-2))/((x+2)(x-2)))(4/(x^2)) + ((x^2(x-2))/(x^2(x-2)))(3/(x+2))}}} Multiply the first term by {{{(x^2(x+2))/(x^2(x+2))}}}. Multiply the second term by {{{((x+2)(x-2))/((x+2)(x-2))}}}. Multiply the third term by {{{(x^2(x-2))/(x^2(x-2))}}}. 



{{{(x^2(x+2))/(x^2(x+2)(x-2)) - (4(x+2)(x-2))/(x^2(x+2)(x-2)) + (3x^2(x-2))/(x^2(x+2)(x-2))}}} Combine the fractions.



{{{(x^3+2x^2)/(x^2(x+2)(x-2)) - (4(x^2-4))/(x^2(x+2)(x-2)) + (3x^2(x-2))/(x^2(x+2)(x-2))}}} FOIL the terms in the numerator.



{{{(x^3+2x^2)/(x^2(x+2)(x-2)) - (4x^2-16)/(x^2(x+2)(x-2)) + (3x^3-6x^2)/(x^2(x+2)(x-2))}}} Distribute.




{{{((x^3+2x^2)- (4x^2-16)+(3x^3-6x^2))/(x^2(x+2)(x-2))}}} Combine the fractions.



{{{(x^3+2x^2- 4x^2+16+3x^3-6x^2)/(x^2(x+2)(x-2))}}} Distribute



{{{(4x^3-8x^2+16)/(x^2(x+2)(x-2))}}} Combine like terms.




So {{{1/(x-2) - 4/(x^2) + 3/(x+2)}}} simplifies to {{{(4x^3-8x^2+16)/(x^2(x+2)(x-2))}}}



In other words, {{{1/(x-2)-4/(x^2)+3/(x+2)=(4x^3-8x^2+16)/(x^2(x+2)(x-2))}}} where {{{x<>-2}}}, {{{x<>0}}}, or {{{x<>2}}}



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# 2




{{{((x^2-6x+8)/(3x+9))((x+3)/(x^2-4))}}} Start with the given expression.



{{{(((x-2)*(x-4))/(3x+9))((x+3)/(x^2-4))}}} Factor {{{x^2-6x+8}}} to get {{{(x-2)*(x-4)}}}.



{{{(((x-2)*(x-4))/(3(x+3)))((x+3)/(x^2-4))}}} Factor {{{3x+9}}} to get {{{3(x+3)}}}.



{{{(((x-2)*(x-4))/(3(x+3)))((x+3)/((x-2)*(x+2)))}}} Factor {{{x^2-4}}} to get {{{(x-2)*(x+2)}}}.



{{{((x-2)*(x-4)(x+3))/(3(x+3)(x-2)*(x+2))}}} Combine the fractions. 



{{{(highlight(x-2)(x-4)highlight(x+3))/((3)highlight(x+3)highlight(x-2)(x+2))}}} Highlight the common terms. 



{{{(cross(x-2)(x-4)cross(x+3))/((3)cross(x+3)cross(x-2)(x+2))}}} Cancel out the common terms. 



{{{(x-4)/(3(x+2))}}} Simplify. 



So {{{((x^2-6x+8)/(3x+9))((x+3)/(x^2-4))}}} simplifies to {{{(x-4)/(3(x+2))}}}.



In other words, {{{((x^2-6x+8)/(3x+9))((x+3)/(x^2-4))=(x-4)/(3(x+2))}}} where {{{x<>-3}}}, {{{x<>-2}}}, or {{{x<>2}}}




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# 3




{{{((y^2+7y+10)/(2y-4))/((y^2-3y-10)/(y-2))}}} Start with the given expression.



{{{((y^2+7y+10)/(2y-4))((y-2)/(y^2-3y-10))}}} Multiply the first fraction {{{(y^2+7y+10)/(2y-4)}}} by the reciprocal of the second fraction {{{(y^2-3y-10)/(y-2)}}}.



{{{(((y+5)*(y+2))/(2y-4))((y-2)/(y^2-3y-10))}}} Factor {{{y^2+7y+10}}} to get {{{(y+5)*(y+2)}}}.



{{{(((y+5)*(y+2))/(2(y-2)))((y-2)/(y^2-3y-10))}}} Factor {{{2y-4}}} to get {{{2(y-2)}}}.



{{{(((y+5)*(y+2))/(2(y-2)))((y-2)/((y+2)*(y-5)))}}} Factor {{{y^2-3y-10}}} to get {{{(y+2)*(y-5)}}}.



{{{((y+5)*(y+2)(y-2))/(2(y-2)(y+2)*(y-5))}}} Combine the fractions. 



{{{((y+5)highlight(y+2)highlight(y-2))/((2)highlight(y-2)highlight(y+2)(y-5))}}} Highlight the common terms. 



{{{((y+5)cross(y+2)cross(y-2))/((2)cross(y-2)cross(y+2)(y-5))}}} Cancel out the common terms. 



{{{(y+5)/(2(y-5))}}} Simplify. 



So {{{((y^2+7y+10)/(2y-4))/((y^2-3y-10)/(y-2))}}} simplifies to {{{(y+5)/(2(y-5))}}}.



In other words, {{{((y^2+7y+10)/(2y-4))/((y^2-3y-10)/(y-2))=(y+5)/(2(y-5))}}} where {{{y<>-2}}}, {{{y<>2}}}, or {{{y<>5}}}