A polynomial f(x) and one or more of its zeros are given. f(x) = x^4 -2x^3 +23x^2 +6x -78. 1=5i is a zero.               a. FInd all zeros. Write the answer in exact form. If there is more than one answer, separate them with a comma

Given the polynomial \( f(x) = x^4 - 2x^3 + 23x^2 + 6x - 78 \) and that \( 1 + 5i \) is a zero, we can use this information to find all the zeros.

### Step 1: Use the complex conjugate theorem
Since \( 1 + 5i \) is a zero, its complex conjugate, \( 1 - 5i \), must also be a zero. Therefore, two of the zeros are:
\[
1 + 5i, \quad 1 - 5i
\]

### Step 2: Form a quadratic factor
Now that we know two of the zeros, we can form the quadratic factor corresponding to these complex roots. The factor is:
\[
(x - (1 + 5i))(x - (1 - 5i))
\]
Using the difference of squares, we simplify:
\[
(x - 1 - 5i)(x - 1 + 5i) = (x - 1)^2 - (5i)^2 = (x - 1)^2 + 25
\]
So the quadratic factor is:
\[
(x - 1)^2 + 25 = x^2 - 2x + 26
\]

### Step 3: Divide the original polynomial by the quadratic factor
Now, we perform synthetic or polynomial division of \( f(x) \) by \( x^2 - 2x + 26 \). Dividing \( f(x) = x^4 - 2x^3 + 23x^2 + 6x - 78 \) by \( x^2 - 2x + 26 \) gives the quotient \( x^2 - 78 \).

### Step 4: Solve the quotient
Now, we solve the quotient \( x^2 - 78 = 0 \):
\[
x^2 = 78
\]
\[
x = \pm \sqrt{78} = \pm \sqrt{3 \times 26} = \pm \sqrt{6 \times 13}
\]

Thus, the remaining two zeros are:
\[
x = \sqrt{78}, \quad x = -\sqrt{78}
\]

### Final Answer:
The four zeros of the polynomial \( f(x) \) are:
\[
1 + 5i, \quad 1 - 5i, \quad \sqrt{78}, \quad -\sqrt{78}
\]

Would you like to see a detailed step-by-step division or have any other questions?

Here are five related questions to think about:
1. How do we use synthetic division with polynomials?
2. What is the complex conjugate theorem, and why is it useful in solving polynomials?
3. How can you confirm that a set of roots is correct for a given polynomial?
4. What is the difference between real and complex zeros of polynomials?
5. How would the graph of \( f(x) = x^4 - 2x^3 + 23x^2 + 6x - 78 \) look with the given roots?

**Tip:** When working with polynomials with complex roots, always remember that the conjugate of a complex number is also a root.

A polynomial f(x) and one or more of its zeros are given. f(x) = x^4 -2x^3 +23x^2 +6x -78. 1+5i is a zero. a. Find all zeros. Write the answer in exact form. If there is more than one answer, separate them with a comma.

Learn how to find all zeros of the polynomial f(x) = x^4 - 2x^3 + 23x^2 + 6x - 78, given that 1+5i is a zero. This problem explores complex numbers, the complex conjugate theorem, synthetic division, and solving polynomials. Suitable for high school students in Grades 10-12.

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