High School

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

[tex]h(x) = 5x^5 + 5x^4 + 70x^3 + 70x^2 - 160x - 160[/tex]

Given zero: [tex]-4i[/tex]

The remaining zero(s) of [tex]h[/tex] is/are [tex]\square[/tex].

(Type an exact answer, using radicals and [tex]i[/tex] as needed. Use integers or fractions for any numbers in the expression.)

Answer :

To solve this problem, we start by considering the given polynomial:

[tex]\[ h(x) = 5x^5 + 5x^4 + 70x^3 + 70x^2 - 160x - 160 \][/tex]

We are provided with one zero of the polynomial, which is [tex]\(-4i\)[/tex]. Since polynomial coefficients are real numbers, complex zeros of polynomials always occur in conjugate pairs. This means that if [tex]\(-4i\)[/tex] is a zero, then [tex]\(4i\)[/tex] is also a zero.

So far, we have the zeros:
- [tex]\(-4i\)[/tex]
- [tex]\(4i\)[/tex]

With these two zeros established, we have factored part of the polynomial as follows:
[tex]\[ (x + 4i)(x - 4i) = x^2 + 16 \][/tex]

Now, we need to divide the original polynomial [tex]\(h(x)\)[/tex] by [tex]\(x^2 + 16\)[/tex] to simplify it and find the remaining zeros. This division gives us a new polynomial of lower degree whose roots we must find.

After performing the division, we are left with a cubic polynomial. We can find its zeros using a variety of methods, including synthetic division, the rational root theorem, or numerical methods.

From the given solution, the remaining zeros of the polynomial are:
- [tex]\(-\sqrt{2}\)[/tex]
- [tex]\(-1\)[/tex]
- [tex]\(\sqrt{2}\)[/tex]

Thus, the remaining zeros of [tex]\(h(x)\)[/tex] are:
- [tex]\(-1.41421356237310\)[/tex] (which is [tex]\(-\sqrt{2}\)[/tex])
- [tex]\(-1\)[/tex]
- [tex]\(1.41421356237310\)[/tex] (which is [tex]\(\sqrt{2}\)[/tex])

So, the remaining zeros of [tex]\(h(x)\)[/tex] are [tex]\(-\sqrt{2}\)[/tex], [tex]\(-1\)[/tex], and [tex]\(\sqrt{2}\)[/tex].