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Answer :
Certainly! Let's find the remaining zeros of the polynomial function [tex]\( h(x) = 2x^4 + 3x^3 + 70x^2 + 108x - 72 \)[/tex], given that one of the zeros is [tex]\(-6i\)[/tex].
### Step 1: Use the Complex Conjugate Theorem
When a polynomial has real coefficients and a complex number is a zero, its complex conjugate is also a zero. Since [tex]\(-6i\)[/tex] is a zero, its conjugate [tex]\(6i\)[/tex] is also a zero.
### Step 2: Find the Remaining Zeros
Since we know two zeros, [tex]\(-6i\)[/tex] and [tex]\(6i\)[/tex], we need to find the remaining zeros of the polynomial. The polynomial is of degree 4, meaning it has four zeros in total.
### Step 3: Summary of Zeros
The polynomial will have the following zeros based on the given:
1. [tex]\(-6i\)[/tex]
2. [tex]\(6i\)[/tex]
3. [tex]\(-2\)[/tex]
4. [tex]\(\frac{1}{2}\)[/tex]
This means the polynomial can be factored with these zeros. Since it is a degree 4 polynomial, we have now found all the zeros. Therefore, the full list of zeros is [tex]\(-6i, 6i, -2, \frac{1}{2}\)[/tex].
Feel free to ask if you need further explanation or details!
### Step 1: Use the Complex Conjugate Theorem
When a polynomial has real coefficients and a complex number is a zero, its complex conjugate is also a zero. Since [tex]\(-6i\)[/tex] is a zero, its conjugate [tex]\(6i\)[/tex] is also a zero.
### Step 2: Find the Remaining Zeros
Since we know two zeros, [tex]\(-6i\)[/tex] and [tex]\(6i\)[/tex], we need to find the remaining zeros of the polynomial. The polynomial is of degree 4, meaning it has four zeros in total.
### Step 3: Summary of Zeros
The polynomial will have the following zeros based on the given:
1. [tex]\(-6i\)[/tex]
2. [tex]\(6i\)[/tex]
3. [tex]\(-2\)[/tex]
4. [tex]\(\frac{1}{2}\)[/tex]
This means the polynomial can be factored with these zeros. Since it is a degree 4 polynomial, we have now found all the zeros. Therefore, the full list of zeros is [tex]\(-6i, 6i, -2, \frac{1}{2}\)[/tex].
Feel free to ask if you need further explanation or details!
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