Factor (x^2+y^2-z^2)^2-4x^2y^2as the product of four polynomials of degree 1.
+2446 I have found a way to factor the original expression (x2+y2−z2)2−4x2y2.
| (x2+y2−z2)2−4x2y2(x2+y2−z2)2−(2xy)2 | Although this may not be immediately obvious, the original expression is a difference of squares. I rewrote the original expression to make this fact clearer. Remember thata2−b2=(a+b)(a−b). Factor the expression as such. I have used colors to make this process easier for your eyes to digest. |
| (x2+y2−z2+2xy)(x2+y2−z2−2xy) | Let's do some rearranging here. |
| (x2+2xy+y2−z2)(x2−2xy+y2−z2) | Why am I highlighting this part? Well, both portions happened to be perfect-square trinomials. |
| ([x+y]2−z2)([x−y]2−z2) | Do you notice something? Yes, it is another difference of squares. It is time to factor again! |
| ([x+y]2−z2)=(x+y+z)(x+y−z)([x−y]2−z2)=(x−y+z)(x−y−z) | I have factored each one individually. The last step is to combine the factors into one expression. |
| (x+y+z)(x+y−z)(x−y+z)(x−y−z) | This expression satisfies the original condition; the final expression is "the product of four polynomials of degree 1." |
+2446 I have found a way to factor the original expression (x2+y2−z2)2−4x2y2.
| (x2+y2−z2)2−4x2y2(x2+y2−z2)2−(2xy)2 | Although this may not be immediately obvious, the original expression is a difference of squares. I rewrote the original expression to make this fact clearer. Remember thata2−b2=(a+b)(a−b). Factor the expression as such. I have used colors to make this process easier for your eyes to digest. |
| (x2+y2−z2+2xy)(x2+y2−z2−2xy) | Let's do some rearranging here. |
| (x2+2xy+y2−z2)(x2−2xy+y2−z2) | Why am I highlighting this part? Well, both portions happened to be perfect-square trinomials. |
| ([x+y]2−z2)([x−y]2−z2) | Do you notice something? Yes, it is another difference of squares. It is time to factor again! |
| ([x+y]2−z2)=(x+y+z)(x+y−z)([x−y]2−z2)=(x−y+z)(x−y−z) | I have factored each one individually. The last step is to combine the factors into one expression. |
| (x+y+z)(x+y−z)(x−y+z)(x−y−z) | This expression satisfies the original condition; the final expression is "the product of four polynomials of degree 1." |
