could someone help me figure this problem out please!!
(a4+4b4)÷ (a2-2ab+2b2)
(a4+4b4)÷ (a2-2ab+2b2)
-
(a^4 + 4b^4) ÷ (a^2 - 2ab + 2b^2)
= [(a^2 - 2ab + 2b^2) (a^2 + 2ab + 2b^2)] / (a^2 - 2ab + 2b^2)
= a^2 + 2ab + 2b^2
Edit: Factoring: a^4 + 4b^4:
First, the formulas(short cuts) that are very useful when factoring:
(a + b)^2 = a^2 + 2ab + b^2 => square of sums
(a - b)^2 = a^2 - 2ab + b^2 => square of deference
and of course one of most important ones:
a^2 - b^2 = (a - b)(a + b) => difference of squares
OK, our problem at first it looks as not factorisable, but sometimes we can use a trick or two to fix that, like adding or subtracting extra terms:
a^4 + 4b^4 => notice that if we had one extra term i.e. 4a^2b^2 , then:
a^4 + 4a^2b^2 + 4b^4 => a^2 + 2ab + b^2 = (a + b)^2, if : a = a^2 , b = 2b^2:
(a^2 + 2b^2)^2 = a^4 + 4a^2b^2 + 4b^4 => well that's great but we can not just add or subtract a value to an expression, we can only do that to both sides of an equation since equal values on both sides would be a null because they cancel each other. So we need to use a neat trick here:
a^4 + 4a^2b^2 + 4b^4 - 4a^2b^2 => the original value is not altered, we get:
(a^2 + 2b^2)^2 - 4a^2b^2 => which is just a difference of squares, then:
(a^2 + 2b^2 - 2ab)(a^2 + 2b^2 + 2ab) => can also be written as:
(a^2 - 2ab + 2b^2) (a^2 + 2ab + 2b^2)
Regards.
= [(a^2 - 2ab + 2b^2) (a^2 + 2ab + 2b^2)] / (a^2 - 2ab + 2b^2)
= a^2 + 2ab + 2b^2
Edit: Factoring: a^4 + 4b^4:
First, the formulas(short cuts) that are very useful when factoring:
(a + b)^2 = a^2 + 2ab + b^2 => square of sums
(a - b)^2 = a^2 - 2ab + b^2 => square of deference
and of course one of most important ones:
a^2 - b^2 = (a - b)(a + b) => difference of squares
OK, our problem at first it looks as not factorisable, but sometimes we can use a trick or two to fix that, like adding or subtracting extra terms:
a^4 + 4b^4 => notice that if we had one extra term i.e. 4a^2b^2 , then:
a^4 + 4a^2b^2 + 4b^4 => a^2 + 2ab + b^2 = (a + b)^2, if : a = a^2 , b = 2b^2:
(a^2 + 2b^2)^2 = a^4 + 4a^2b^2 + 4b^4 => well that's great but we can not just add or subtract a value to an expression, we can only do that to both sides of an equation since equal values on both sides would be a null because they cancel each other. So we need to use a neat trick here:
a^4 + 4a^2b^2 + 4b^4 - 4a^2b^2 => the original value is not altered, we get:
(a^2 + 2b^2)^2 - 4a^2b^2 => which is just a difference of squares, then:
(a^2 + 2b^2 - 2ab)(a^2 + 2b^2 + 2ab) => can also be written as:
(a^2 - 2ab + 2b^2) (a^2 + 2ab + 2b^2)
Regards.