Answer :
Answer:
[tex]f(x)=x^{4}+x^{3}-10x^{2} +8x[/tex]
Step-by-step explanation:
A number is a factor of f(x) if and only if f(x) is zero for that value/number.
For the factors of a function we write the factors as x-a where a is the zero of function i.e. value at which f(x) is zero.
To write the polynomial function of minimum degree with real coefficients whose zeros include 2, -4, and 1, 3i, we find the f(x) is the product of all factors i.e x-a where a will represent the given zeros.
[tex]f(x)=(x-2)(x-(-4))(x-1)(x-3i)\\f(x)=(x-2)(x+4))(x-1)(x-3i)\\f(x)=(x^{2} +4x-2x-8)(x^{2} -3xi-x+3i )\\f(x)=(x^{2} +2x-8)(x^{2} -x-(3x+3)i)\\[/tex]
As it is stated that polynomial should have real coefficients so skipping the terms with 'i' we get
[tex]f(x)=(x^{2} +2x-8)(x^{2} -x)\\f(x)=x^{4}-x^{3}+2x^{3}-2x^{2} -8x^{2} +8x\\f(x)=x^{4}+x^{3}-10x^{2} +8x[/tex]
