How do I completely factor this polynomial? (x^2+1)^2 -4x^2?

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A few days ago

math guy

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Anytime you have the difference of perfect squares (like in your problem), they can be factored as follows:

a^2 – b^2

= (a – b)(a + b)

In your problem, a = (x^2+1) and b = 2x.

(Yes, b = 2x, not 4x since 2x*2x = 4x^2.)

So (x^2+1)^2 -4x^2

= ((x^2+1) – 2x)((x^2+1) + 2x)

= (x^2 – 2x + 1)(x^2 + 2x + 1)

Now both of these are called “perfect square trinomials” because they are the squares of monomials:

= (x-1)(x-1) ( x+1)(x+1)

= (x-1)^2 * (x+1)^2

I hope this helps!

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