Answer:
Third option: [tex](3 + xz)(-3 + xz)[/tex]
Fourth option: [tex](y^2 - xy)(y^2 + xy)[/tex]
Sixth option: [tex](64y^2 + x^2)(-x^2 + 64y^2)[/tex]
Step-by-step explanation:
By definition, we can factor the Difference between two squares:
[tex]a^2 - b^2 = (a+b)(a-b)[/tex]
In order to find which products results in a difference of squares, we need to check each option:
[tex]1.\ (x - y)(y - x)=(x)(y)+(x)(-x)-(y)(y)+(-y)(-x)=xy-x^2-y^2+xy=-x^2-y^2+2xy[/tex]
The product does not result in a difference of squares.
[tex]2.\ (6 - y)(6 - y)[/tex]
Since the signs are equal ([tex]-[/tex]), the product will not result in a difference of squares.
Using Distributive Property for the other options, we get:
[tex]3.\ (3 + xz)(-3 + xz)=(3)(-3)+(3)(xz)+(xz)(-3)+(xz)(xz)=\\\\=-9+3xz-3xz+x^2z^2=x^2z^2-9=(xz)^2-(3)^2[/tex]
The product results in a difference of squares.
[tex]4.\ (y^2 - xy)(y^2 + xy)=(y^2)(y^2)+(y^2)(xy)-(xy)(y^2)-(xy)(xy)=\\\\=y^4+xy^3-xy^3-x^2y^2=y^4-x^2y^2=(y^2)^2-(xy)^2[/tex]
The product results in a difference of squares.
[tex]5.\ (25x - 7y)(-7y + 25x)=(25x - 7y)(25x - 7y)[/tex]
Since the signs are equal ([tex]-[/tex]), the product will not result in a difference of squares.
[tex]6.\ (64y^2 + x^2)(-x^2 + 64y^2)=(64y^2 + x^2)(64y^2 - x^2)=\\\\=(64y^2)(-x^2)+(64y^2)(64y^2)+(x^2)(-x^2)+(x^2)(64y^2)=\\\\=-64y^2x^2+4096y^4-x^4+64y^2x^2=\\\\=4096y^4-x^4=(64y^2)^2-(x^2)^2[/tex]
The product results in a difference of squares.
