Question

What is The polynomial for the question above

Answer 1

A polynomial of degree n with roots $r_1,r_2,...,r_n$ can be represented as: $f(x)=A(x-r_1)(x-r_2)\cdot...\cdot(x-r_n)$, where A is a constant (The same "a" that appears outside the parentheses in the field where you will type the polynomial). In this question, the number of roots is 4, because the degree of f(x) is 4. If a polynomial has only real coefficients, the complex roots appear in pairs of conjugates. So, since $-3+4i$ is a zero, $-3-4i$ is a zero too. Then, we have the 4 zeros. Hence, $f(x)$ will be in the form:

$f(x)=A(x-5)^2(x-(-3+4i))(x-(-3-4i))\\\\f(x)=A(x-5)^2(x^2-(-3+4i)x-(-3-4i)x+(-3+4i)(-3-4i))\\\\f(x)=A(x^2-10x+25)(x^2+6x+25)\\\\f(x)=A(x^4-4x^3-10x^2-100x+625)$