Simplify each of the following expressions into simplest terms.
a) {[-X+Y+(-2)*(-2Y)*[X-Y]}/[(X-Y)^2+(X+Y)^2]
b)
(X^3+4X^2Y+4XY^2+2XY-6Y^2)/(X+2Y)
Please don't capitalize your variables, it makes it hard to read them.
(-x + y - 2(-2y)(x - y) )/( (x-y)^2 + (x+y)^2 )
= (-x + y + 4xy - 4y^2)/(x^2 - 2xy + y^2 + x^2 + 2xy + y^2)
= ( x(4y - 1) - y(4x - 1) )/( 2(x^2 + y^2) )
= (4x-1)(x-y) / (2(x^2 + y^2) )
for the second, I don't see much happening
you could separate the fraction into
(x^3 + 4x^2y + 4xy^2)/(x+2y) + (2xy - 6y^2)/(x+2y)
= x(x^2 + 4xy + 4y^2)/(x+2y) + 2y(x - 3y)/(x-2y)
= x(x-2y)^2 / (x-2y) + 2y(x-3y)/(x-2y
= x(x-2y) + 2y(x-3y)/(x-2y)
if that second-last term had been 3xy we could have done more.
Was there a typo ?
no no typo the equation is
(x^3+4x^2y+4xy^2+2xy-6y^2)/(x+2y)
sorry about the caps on my first post
i did however seem to have made a typo on the first equation the right one is
{[-x+y+(-2y)]*[x-y]}/[(x-y)^2+(x+y)^2]
If no typo in the second, then my answer stands as is
for the first, follow my steps with your corrected expression. There should be only minor changes.
To simplify the given expressions, we'll follow the standard rules of simplification for algebraic expressions.
a) Let's simplify the expression {[-X+Y+(-2)*(-2Y)*[X-Y]}/[(X-Y)^2+(X+Y)^2]
1. Remove the parentheses within the brackets:
[-X+Y-4Y*[X-Y]]/[(X-Y)^2+(X+Y)^2]
2. Simplify the multiplication inside the brackets:
[-X+Y-4Y(X-Y)]/[(X-Y)^2+(X+Y)^2]
3. Expand the multiplication:
[-X+Y-4YX+4Y^2]/[(X-Y)^2+(X+Y)^2]
4. Combine like terms:
[-X-4YX+Y+4Y^2]/[(X-Y)^2+(X+Y)^2]
This is the simplified expression.
b) Let's simplify the expression (X^3+4X^2Y+4XY^2+2XY-6Y^2)/(X+2Y)
1. Start with long division:
Divide X^3+4X^2Y+4XY^2+2XY-6Y^2 by X+2Y
X^2 - 2XY + 6Y^2
_______________________
X +2Y | X^3 + 4X^2Y + 4XY^2 + 2XY - 6Y^2
2. Divide X^3 by X: X^2
Multiply X^2 by X+2Y: X^3 + 2XY^2
3. Subtract (X^3 + 2XY^2) from (X^3 + 4X^2Y + 4XY^2):
-2XY^2 + 2XY + 2X^2Y
4. Bring down the next term: 2X^2Y
5. Divide 2X^2Y by X: 2XY
Multiply 2XY by X+2Y: 2X^2Y + 4XY^2
6. Subtract (2X^2Y + 4XY^2) from (-2XY^2 + 2XY + 2X^2Y):
-2XY^2 + 2XY + 2X^2Y - 2X^2Y - 4XY^2 = -6XY^2
7. Bring down the next term: -6XY^2
8. Divide -6XY^2 by X: -6Y^2
Multiply -6Y^2 by X+2Y: -6XY^2 - 12Y^3
9. Subtract (-6XY^2 - 12Y^3) from (-6XY^2):
-6XY^2 - 12Y^3
10. Since there is nothing left to bring down, we are finished with long division.
The simplified expression is X^2 - 2XY + 6Y^2 - 6Y^2/(X+2Y), which can be further simplified to X^2 - 2XY.