Form a fifthdegree polynomial function with real coefficient

Form a fifth-degree polynomial function with real coefficients such that 5i, 1 - 3i and -1 are zeros and f(0) = 750. f(x) = (Simplify your answer. Type an expression using x as the variable.)

Solution

solution

Real coefficients means complex roots come in complex conjugate pairs.

f(x)= a(x-5i)(x+5i)(x-(1-3i)(x-(1+3i)))(x+1)

f(x)= a(x^2+25i^2)(x-1+3i)(x-1-3i)(x+1)

f(x)= a(x^3+x^2-25x-25)(x^2-x-3xi-x+1+3i+3xi-3i+9)

f(x)= a(x^3+x^2-25x-25)(x^2-2x+10)

x=0

f(0)=750

750= a(-25+10)

750= -15a

a= -50

f(x)= -50(x^3+x^2-25x-25)(x^2-2x+10)

f(x)= -50(x^5-2x^4+10x^3+x^4-2x^3+10x^2-25x^3+50x^2-250x-25x^2+50x-250)

f(x)= -50(x^5-x^4-17x^3+35x^2-200x-250)