Because the polynomial function has a degree of 5, its general form will be f(x)=a(x-x_1)(x-x_2)(x-x_3)(x-x_4)(x-x_5). In this form a is the leading coefficient and x_1, x_2, x_3, x_4, and x_5 are the zeros of the function.
Possible Function: f(x)=(x-i)(x+i)(x-sqrt(5))(x+sqrt(5))(2x-1)
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Because the polynomial function has a degree of 5, its general form will be as shown below.
f(x)=a(x-x_1)(x-x_2)(x-x_3)(x-x_4)(x-x_5)
In this form a is the leading coefficient and x_1, x_2, x_3, x_4, and x_5 are the zeros of the function. We want two imaginary zeros and two irrational zeros. Possible zeros can be listed as below.
Imaginary Zeros:& x_1=i and x_2=- i
Irrational Zeros:& x_3=sqrt(5) and x_4=- sqrt(5)
We also need one non-integral zero. A non-integral rational number is a fraction where numerator and denominator cannot be canceled to leave an integer value but stay a real fraction. Therefore, one possible non-integral zero can be the following.
Non-integral Zero: x_5=1/2
Let the leading coefficient be a=2. Now, we are ready to write the function.
f(x)=2(x-i)(x+i)(x-sqrt(5))(x+sqrt(5))(x-1/2)
⇓
f(x)=(x-i)(x+i)(x-sqrt(5))(x+sqrt(5))(2x-1)
