Given the zeros of the function:
[tex]\begin{gathered} 6 \\ 7i \\ -7i \end{gathered}[/tex]
You can write the equation in Factored Form:
[tex]f(x)=\mleft(x-6\mright)\mleft(x-7i\mright)\mleft(x+7i\mright)[/tex]
Now you need to simplify:
1. Remember this formula:
[tex](a+b)(a-b)=a^2-b^2[/tex]
In this case:
[tex]a=x[/tex][tex]b=7i[/tex]
Therefore, you can rewrite the expression in this form:
[tex]f(x)=(x-6)((x)^2-(7i)^2)[/tex]
2. By definition:
[tex]\begin{gathered} i=\sqrt[]{-1} \\ \\ i^2=-1 \end{gathered}[/tex]
Then:
[tex]\begin{gathered} f(x)=(x-6)(x^2-49(-1)) \\ \\ f(x)=(x-6)(x^2+49) \end{gathered}[/tex]
3. Now you need to use the FOIL Method in order to multiply the binomials. This states that:
[tex](a+b)\mleft(c+d\mright)=ac+ad+bc+bd[/tex]
Hence:
[tex]\begin{gathered} f(x)=(x)(x^2)+(x)(49)-(6)(x^2)-(6)(49) \\ \\ f(x)=x^3+49x-6x^2-294 \end{gathered}[/tex]
4. Ordering the polynomial from the highest power to the least power, you get:
[tex]f(x)=x^3-6x^2+49x-294[/tex]
Hence, the answer is: Option d.
