Answer :
To find the remaining zeros of the polynomial function given one zero as [tex]\(-6i\)[/tex], we can use the property of polynomials that states complex roots occur in conjugate pairs. Therefore, if [tex]\(-6i\)[/tex] is a zero, its conjugate [tex]\(6i\)[/tex] is also a zero.
We start with the function:
[tex]\[ h(x) = 7x^5 + 2x^4 + 210x^3 + 60x^2 - 1512x - 432 \][/tex]
1. Identify Known Zeros:
The given zero is [tex]\(-6i\)[/tex]. This implies [tex]\(6i\)[/tex] is also a zero due to the conjugate pair property.
2. Search for Other Zeros:
The remaining zeros can be found by dividing the polynomial by the quadratic factor associated with the complex zeros [tex]\((-6i)\)[/tex] and [tex]\(6i\)[/tex], which is [tex]\(x^2 + 36\)[/tex].
3. Factor and Solve:
We can then factor the polynomial or use other methods to find the remaining zeros. The factorization reveals that the remaining zeros are:
- [tex]\(-\frac{2}{7}\)[/tex]
- [tex]\(-\sqrt{6}\)[/tex]
- [tex]\(\sqrt{6}\)[/tex]
4. Conclusion:
Thus, the remaining zeros of the function [tex]\(h(x)\)[/tex] are [tex]\(-\frac{2}{7}\)[/tex], [tex]\(-\sqrt{6}\)[/tex], and [tex]\(\sqrt{6}\)[/tex].
These are the additional roots, apart from the ones initially given. If you have any more questions or need further clarification, feel free to ask!
We start with the function:
[tex]\[ h(x) = 7x^5 + 2x^4 + 210x^3 + 60x^2 - 1512x - 432 \][/tex]
1. Identify Known Zeros:
The given zero is [tex]\(-6i\)[/tex]. This implies [tex]\(6i\)[/tex] is also a zero due to the conjugate pair property.
2. Search for Other Zeros:
The remaining zeros can be found by dividing the polynomial by the quadratic factor associated with the complex zeros [tex]\((-6i)\)[/tex] and [tex]\(6i\)[/tex], which is [tex]\(x^2 + 36\)[/tex].
3. Factor and Solve:
We can then factor the polynomial or use other methods to find the remaining zeros. The factorization reveals that the remaining zeros are:
- [tex]\(-\frac{2}{7}\)[/tex]
- [tex]\(-\sqrt{6}\)[/tex]
- [tex]\(\sqrt{6}\)[/tex]
4. Conclusion:
Thus, the remaining zeros of the function [tex]\(h(x)\)[/tex] are [tex]\(-\frac{2}{7}\)[/tex], [tex]\(-\sqrt{6}\)[/tex], and [tex]\(\sqrt{6}\)[/tex].
These are the additional roots, apart from the ones initially given. If you have any more questions or need further clarification, feel free to ask!
