Factoring Problem

[Galasriniel10]

Factor completely

x^4-28x^2+75

 

[tkhunny]

First step, what would you do it if it were this: z^2 - 28z + 75 ?

 

[BigGlenntheHeavy]

\(\displaystyle f(x) \ = \ x^4-28x^2+75, \ now \ what \ times \ what \ = \ 75 \ and \ adds \ to \ -28? \ How \ about \ -3 \ and \ -25.\)

\(\displaystyle Ergo, \ by \ grouping, \ we \ have: \ x^4-25x^2-3x^2+75 \ = \ f(x), \ \implies \ f(x) \ = \ x^2(x^2-25)-3(x^2-25).\)

\(\displaystyle Hence, \ we \ now \ have \ f(x) \ = \ (x^2-3)(x^2-25) \ = \ (x-\sqrt3)(x+\sqrt3)(x-5)(x+5), \ QED\)

 

[Galasriniel10]

Thank you!!!