Answer:
[tex]P(x)=x^3-ix^2-4x+4i[/tex]
Step-by-step explanation:
We have to find the polynomial of lowest degree with lead coefficient 1 and roots i, -2 and 2.
A polynomial can be written as:
[tex]P(x)=a*(x-x_1)*(x-x_2)*...*(x-x_n)[/tex]
Where [tex]a[/tex] is the lead coefficient. And [tex]x_1,x_2,...,x_n[/tex] are the roots of the polynomial.
Then we have [tex]a=1[/tex] and,
[tex]x_1=i\\x_2=-2\\x_3=2[/tex]
We can write the polynomial as:
[tex]P(x)=1(x-i)(x-(-2))(x-2)\\P(x)=(x-i)(x+2)(x-2)[/tex]
You can apply squared binomial to [tex](x+2)(x-2)[/tex]:
[tex](x+2)(x-2)=x^2-2^2=(x^2-4)[/tex]
Then,
[tex]P(x)=(x-i)(x^2-4)[/tex] apply distributive property:
[tex]P(x)=(x-i)(x^2-4)\\P(x)=x^3-4x-ix^2+4i\\P(x)=x^3-ix^2-4x+4i[/tex]
The the polynomial of lowest degree with leaf coefficient 1 and roots i, -2 and 2 is:
[tex]P(x)=x^3-ix^2-4x+4i[/tex]
