Factor these polynomial. (See Examples 1 and 2.)x4 − x2 − 12Example 1Factor the following...

Factor these polynomial. (See Examples 1 and 2.)

x4 − x2 − 12

Example 1

Factor the following polynomials.

(a) x8 + 4x4 + 3

Solution The idea is to make a substitution that reduces the polynomial to a quadratic or cubic that we can dead with. Note that x8 = (x4)2. Let u = x4. Then

 

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(b) x4 − y4

Solution Note that x4 = (x2)2, and similarly for the y term. Let u = x2 and v = y2. Then

 

Example 2

Factor 256k4 − 625m4.

Solution Use the difference of squares twice, as follows: