Final answer:
To factor the expression x^4 - 5x^2 + 4 completely, you can use the substitution a = x^2 to simplify the expression. Then, factor the quadratic expression (a^2 - 5a + 4) into (a - 4)(a - 1), and substitute a = x^2 back in to get (x^2 - 4)(x^2 - 1). Finally, factor the difference of squares to get the completely factored form (x - 2)(x + 2)(x - 1)(x + 1).
Step-by-step explanation:
To factor the expression x^4 - 5x^2 + 4 completely, we will substitute x^2 with a temporary variable. Let a = x^2.
Using this substitution, the expression becomes a^2 - 5a + 4. We can then factor this quadratic expression into (a - 4)(a - 1).
Now, substitute a = x^2 back into the factored expression to get (x^2 - 4)(x^2 - 1).
Next, we can factor the difference of squares: (x^2 - 4) = (x^2 - 2^2) = (x - 2)(x + 2), and (x^2 - 1) = (x - 1)(x + 1).
So, the completely factored form of the expression x^4 - 5x^2 + 4 is (x - 2)(x + 2)(x - 1)(x + 1).