A polynomial P is given.P(x) = x5 − 625x(a) Factor P into linear and irreducible quadratic factors with real coefficients.

The given polynomial is P(x) = x^5 - 625x.

First, we can factor out an x from each term to simplify the polynomial:

P(x) = x(x^4 - 625).

Next, we recognize that x^4 - 625 is a difference of squares, which can be factored as follows:

x^4 - 625 = (x^2 - 25)(x^2 + 25).

Again, we see that x^2 - 25 is a difference of squares, which can be factored as follows:

x^2 - 25 = (x - 5)(x + 5).

However, x^2 + 25 cannot be factored further with real coefficients because it would involve the square root of a negative number.

So, the polynomial P(x) factored into linear and irreducible quadratic factors with real coefficients is:

P(x) = x(x - 5)(x + 5)(x^2 + 25).