P(x) = x^3 − 5x^2 + 9x − 45 (a) Factor P into linear and irreducible quadratic factors with real coefficients. P(x) = ? (b) Factor P completely into linear factors with complex coefficients. P(x)= ?

Answer 1

Look at how x-5 divides the first two and second two terms.

This gives you the answer to the first part right off

P(x) = (x-5)(x² + 9) The second term, of course, has two imaginary roots, so for real coefficients, we stop here.

For the second part, where complex coefficients are allowed, this becomes

P(x) = (x-5)(x+3i)(x-3i)