Question

find the polynomial f(x) with real coefficients having the degrees and zeros
degree 4; zeros -3+5i; -5 multiplicty 2
enter the polynomial

Answer 1

i know the beginning is (n)x^4.

Answer 2

A degree 4 polynomial will have the form
\[ax^4+bx^3+cx^2+dx+e\]
and given its 4 roots, you can write it in its factored form,
\[(x-r_1)(x-r_2)(x-r_3)(x-r_4)\]
where \(r_1,r_2,r_3,r_4\) denote the roots (not necessarily distinct).

You know that \(x=-3+5i\) is a root. Since it is complex, you also know that its conjugate, \(x=-3-5i\) is a root as well (complex roots come in conjugate pairs for polynomials with real coefficients).

You're also given that \(x=-5\) is a root of multiplicity 2, which means \((x+5)^2\) is a factor.

So, your polynomial would be
\[(x-(-3+5i))(x-(-3-5i))(x-(-5))^2\]
Expand and you'll get your answer.