Taking into acount the square of a sum and common factor, you obtain that
(x²+y²)² + 2x²y² - y⁴= x²(x² + 4y²)
The square of a sum is the sum of the squares plus twice the product. That is, the result of the square of a sum of two terms is equal to the square of the first plus twice the product of the first and the second, plus the square of the second.
Calling the terms of the sum "a" and "b", mathematically that is expressed by:
(a+b)²= (a+b)×(a+b)= a×a + a×b + b×a + b×b= a² + 2a×b + b²
In this case, you can express (x²+y²)², being a=x² and b=y², as:
(x²+y²)²= (x²)² + 2x²y² + (y²)²
(x²+y²)²= x⁴ + 2x²y² + y⁴
So, the expression (x²+y²)² 2x²y² - y⁴ can be written as:
(x²+y²)² + 2x²y² - y⁴= x⁴ + 2x²y² + y⁴ + 2x²y² - y⁴
(x²+y²)² + 2x²y² - y⁴= x⁴ + 4x²y²
On the other side, the common factor is the number or variable found in all the terms of an expression.
If a common factor appears in all the terms of an expression, this expression can be written as the product of that factor by the expression that results from dividing each term by that factor.
It should be taken into account that when a factor with different exponents is repeated, then the one with the lowest exponent must be considered as a common factor.
In this case, the common factor is x². So, finally the expression (x²+y²)² 2x²y² - y⁴ can be written as:
(x²+y²)² + 2x²y² - y⁴= x⁴ + 4x²y²= x²(x² + 4y²)
(x²+y²)² + 2x²y² - y⁴= x²(x² + 4y²)
Learn more about common factor and square of a sum:
