\[ P(x)=x^{5}-625 x \] (a) Factor \( P \) into linear and irreducible quadratic factors with real coefficients. \[ P(x)=\square \] (b) Factor \( P \) completely into linear factors with complex coefficients. \( P(x)= \)

To factor \( P(x) = x^5 - 625x \), we begin by factoring out the greatest common factor, which is \( x \). Thus, we can write: \[ P(x) = x(x^4 - 625) \] Next, \( x^4 - 625 \) can be recognized as a difference of squares: \[ x^4 - 625 = (x^2 - 25)(x^2 + 25) \] The term \( x^2 - 25 \) can be factored further as another difference of squares: \[ x^2 - 25 = (x - 5)(x + 5) \] Now, \( x^2 + 25 \) is an irreducible quadratic factor over the real numbers because it cannot be factored further using real coefficients. Thus, we have: \[ P(x) = x(x - 5)(x + 5)(x^2 + 25) \] (a) Therefore, the factorization into linear and irreducible quadratic factors with real coefficients is: \[ P(x) = x(x - 5)(x + 5)(x^2 + 25) \] (b) Now let's explore the factorization into linear factors with complex coefficients. The factor \( x^2 + 25 \) can be factored further using complex numbers: \[ x^2 + 25 = x^2 + (5i)^2 = (x - 5i)(x + 5i) \] Putting it all together, the complete factorization of \( P(x) \) into linear factors with complex coefficients is: \[ P(x) = x(x - 5)(x + 5)(x - 5i)(x + 5i) \]