# Can you help me again with algebra? thanks!?

Help again, with explanation preferably!

A degree 4 polynomial P(x) with integer coefficients has zeros 5i and 1, with 1 being a zero of multiplicity 2. Moreover, the coefficient of x^4 is 1. Find the polynomial. Note: The polynomial must be expanded so that no imaginary number i appears in the polynomial.

College algebra! Thanks!!

Update:

uhm...

Update 2:

what does that mean??? about the black hair?? is that telling me im smart lol? well the answer is: because I was absent that day lol

Relevance

1 being a zero of multiplicity 2. so (x-1)^2 is a facttor

5i is z zero and integer(or real coefficient) so (x-5i) and (x+5i) are 2 factors

so p(x) = A (x-1)^2(x-5i)(x+5i) = A (x-1)^2(x^2+25) where A is a constant

coefficent of x^4 is one so A = 1

so p(x) = (x-1)^2(x^2+ 25) = (x^2-2x + 1)(x^2+25) = x^4 -2x^3+26x^2-50x + 25

• If the polynomial is of degree 4, the number of complex roots (real and imaginary roots) it has should be 4. Since we already have 5i, and two 1's (because it is of multiplicity 2), then we already have 3 roots. However, irrational and imaginary roots always occur in pairs as conjugates. That is, if you have a root of 5i, then you must have a root of -5i also. This completes our 4 roots. Thus, we have

f(x) = A(x - 1)^2 (x + 5i) (x - 5i)

Expanding, we have

f(x) = A (x^2 - 2x + 1)(x^2 + 25)

f(x) = A (x^4 - 2x^3 + 26x^2 - 50x + 25)

Since the coefficient of x^4 is 1, then A must also be 1. Note that the A in front is used to make sure that you get any common factor the original polynomial ever had. Since A will be the coefficient of x^4 when distributed, then it should be 1.

Thus, the polynomial is

f(x) = x^4 - 2x^3 + 26x^2 - 50x + 25

Hope this helps!!!!!

• Two things:

What the hell does this mean?