perform the indicated operation and if possible simplify. (2x+4y)/(2x^2+5xy+2y^2)*(4x^2-y^2)/(8x^2-8)÷(x^2+4xy+4y^2)/(x^2-6xy+9y^2)
i don't understand this the answer is (2x-y)(x-3y)^2/4(x-1)(x+1)(x+2y)^2
It factors to
2(x+2y)/((2x+1)(x+2)) * (2x-y)(2x+y)/((8(x+1)(x-1)) * (x-3y)(x-3y)/((+2y)(x+2y)) , I changed the division at the end to multiplication by "flipping" the fraction
= (2x-y)(x-3y)^2 / [ 4(x+1)(x-1)(x+2y)^2 ]
= their answer
To simplify the given expression, let's break it down step by step.
Step 1: Start by multiplying the numerator and denominator of the main fraction to simplify it:
(2x+4y)/(2x^2+5xy+2y^2) * (4x^2-y^2)/(8x^2-8)
Step 2: Next, we have a division operation, so we can rewrite it as multiplying by the reciprocal of the denominator:
(2x+4y)/(2x^2+5xy+2y^2) * (4x^2-y^2)/(8x^2-8) * (x^2-6xy+9y^2)/(x^2+4xy+4y^2)
Step 3: Now, let's factorize the expressions to find common factors that can be canceled out. We can factorize the denominator expressions as follows:
2x^2 + 5xy + 2y^2 = (2x+y)(x+2y)
8x^2 - 8 = 8(x^2 - 1) = 8(x-1)(x+1)
x^2 + 4xy + 4y^2 = (x+2y)^2
x^2 - 6xy + 9y^2 = (x-3y)^2
Step 4: With the factorization done, we can cancel out common factors between the numerator and denominator:
(2x+4y)/(2x+y)(x+2y) * (4x-y)(x-3y)^2 / 8(x-1)(x+1) * (x-3y)^2 / (x+2y)^2
Step 5: Now, simplify the expression:
(2x+4y)/(8(x-1)(x+1)) * (4x-y)/(x+2y)^2 * (x-3y)^2
Step 6: Finally, multiply the numerators together and the denominators together to get the simplified expression:
(2x+4y)(4x-y)(x-3y)^2 / 8(x-1)(x+1)(x+2y)^2
So, the simplified form of the given expression is (2x+4y)(4x-y)(x-3y)^2 / 8(x-1)(x+1)(x+2y)^2, which matches the answer you provided: (2x-y)(x-3y)^2/4(x-1)(x+1)(x+2y)^2.