Factor x^4-10x^2+9 ?

Question

hi, I need some help factoring that equation ^ 

Answer 1

Since the first term of the expression has a degree of 4 and the middle term has a degree of 2, we can factor using FOIL.  The binomials factors must each have a degree of 2 because when you perform the outer and inner terms, their sum (middle term) will have a degree of 2.  Also, the first terms when expanded out will be a degree of 4.

 

Factoring using FOIL give us

 

(x² - 9)(x² - 1)

 

 

Then notice that each binomial factor is a difference of perfect squares.

 

(x + 3)(x - 3)(x + 1)(x - 1)

Answer 2

x⁴ - 10x² + 9 = (x²)² - 10(x²) + 9

 

Let u = x²

 

Then, the polynomial can be rewritten as u² - 10u + 9

 

                                                     = (u - 9)(u - 1)

 

                                                     = (x² - 9)(x² - 1)

 

                                                     = (x + 3)(x - 3)(x - 1)(x + 1)