What is the full equation including P(x)=?

Question

Given , and that  is a zero, write  in factored form (as a product of linear factors). Be sure to write the full equation, including .

Answer 1

Since 5i is a root and all the coefficients are real, then -5i must also be a root.

Since both 5i and -5i are roots, then P(x) must be divisible by both (x - 5i) and (x + 5i), and so it must be divisible by the product: (x - 5i) (x + 5i) = (x² + 25).

Also, one should always check for roots at 1 and -1, where we find that P(-1) = 0, and so (x+1) is also a factor.

Dividing out (x² + 5) and (x + 1) leaves us with the quadratic: 3x² - 4x - 15, whose roots are easily found to be 3 and -5/3, and so the quadratic can be factored as (x - 3) (3x + 5)

Hence, in factored form, P(x) = (x - 5i) (x + 5i) (x + 1) (x - 3) (3x + 5)