The degree of a polynomial is the greatest exponent in the polynomial. In this case, we are told that the degree is 2.
The leading coefficient is the coefficient of the term with the highest power. We are told that the leading coefficient is -2.
The zeros of a polynomial are the values of x that make the polynomial zero. We are told that the zeros are 7i and -7i.
We can find the polynomial by taking each zero and using it as a root in a factor of the polynomial. In other words, if r is a root (or zero), then (x - r) is a factor.
Using these rules, let's construct our polynomial:
Take the first zero (7i) and make it a factor: (x - 7i)
Do the same for the second zero (-7i). Now you have another factor: (x + 7i)
Combine the two factors. Remember the rule: if r is a root, then (x - r) is a factor. Put these factors together:
(x - 7i)(x + 7i)
Now, recall that the leading coefficient of the polynomial is -2. Therefore, you should multiply the whole expression by -2 to get the complete polynomial:
-2(x - 7i)(x + 7i)
So the polynomial f(x) that meets these requirements is -2(x - 7i)(x + 7i). This is also the complete factored form of the polynomial.
Finally, you can simplify this polynomial by multiplying the factors together to calculate the expanded form of the polynomial:
-2[(x - 7i)(x + 7i)] simplifies to -2x^2 + 98.