Question

Factor the following polynomial completely -x2 y2 + x4 + 4y2 -4x2

Answer 1

The factors of
`-x^(2) y^(2)+x^(4)+4 y^(2)-4 x^(2) \text { are }(2-x),(2+x),(y-x),(y+x)`

SOLUTION:

Given, polynomial is
`-x^(2) y^(2)+x^(4)+4 y^(2)-4 x^(2)`

This is an polynomial in two variables with degree 4

So the given polynomial will have 4 factors.

We need to factorise the given polynomial.

Now,
`\begin{array}{l}{-x^(2) y^(2)+x^(4)+4 y^(2)-4 x^(2)} \\ {\left(-x^(2) y^(2)+4 y^(2)\right)+\left(x^(4)-4 x^(2)\right)}\end{array}`

[writing terms with y as one part and remaining as another part]\

Taking the common terms out of brackets.

`y^(2)\left(-x^(2)+4\right)+-x^(2)\left(-x^(2)+4\right)`

Taking
`\left(-x^(2)+4\right)` as common

`\left(2^(2)-x^(2)\right)\left(y^(2)-x^(2)\right)`

`(2-x)(2+x)(y-x)(y+x)\left[a^(2)-b^(2)=(a-b)(a+b)\right]`

Hence the factors of
`-x^(2) y^(2)+x^(4)+4 y^(2)-4 x^(2) \text { are }(2-x),(2+x),(y-x),(y+x)`