Question

Use the given zero to find the remaining zeros of the function.

h(x)=3x^4+10x³+19x^2+90x-72; zero: -3i

The remaining zeros of h are ___

(Use a comma to separate answers as needed. Use integers or fractions for any numbers in the expression.)

Answer 1

Since the coefficients of the polynomial function h(x) have real values, if -3i is a zero of h(x), then its complex conjugate 3i must also be a zero of h(x). This is known as the Complex Conjugate Root Theorem.

Therefore, the remaining zeros of h(x) can be found by dividing the polynomial by the factors (x - (-3i)) and (x - 3i), using either polynomial long division or synthetic division. Here, we'll use synthetic division:

3i | 3 10 19 90 -72

| 9i -63 -216 -294

+-----------------------

3 10+9i -44 -126 -366

Explanation:

The quotient is 3x^3 + (10+9i)x^2 - 44x - 126, and the remainder is -366. This tells us that:

h(x) = (x - (-3i))(x - 3i)(3x^3 + (10+9i)x^2 - 44x - 126)

The remaining zeros of h(x) can now be found by solving the cubic polynomial 3x^3 + (10+9i)x^2 - 44x - 126. This can be done using various methods such as the Rational Root Theorem, factoring by grouping, or numerical methods. However, these methods can be quite involved and may not always lead to exact answers.