Find all zeros of the function ​f(x)equals=left parenthesis x squared plus 5 x minus 6 right parenthesis squared left parenthesis x minus 9 plus StartRoot 7 EndRoot right parenthesis Superscript 4x2+5x−62x−9+74 and their multiplicities.

We are tasked with finding the zeros of the function:

\[
f(x) = \left( x^2 + 5x - 6 \right)^2 \left( x - 9 + \sqrt{7} \right)^4
\]

### Step 1: Zeros of \( \left( x^2 + 5x - 6 \right)^2 \)
The expression \( \left( x^2 + 5x - 6 \right) \) is a quadratic function. To find the zeros, we solve the quadratic equation:

\[
x^2 + 5x - 6 = 0
\]

We can factor this equation as follows:

\[
x^2 + 5x - 6 = (x + 6)(x - 1)
\]

Thus, the zeros are:
- \( x = -6 \)
- \( x = 1 \)

Since this quadratic is squared in the original function, the multiplicity of both zeros is 2.

### Step 2: Zeros of \( \left( x - 9 + \sqrt{7} \right)^4 \)
The second factor is \( \left( x - 9 + \sqrt{7} \right)^4 \). The zero of this factor occurs when:

\[
x - 9 + \sqrt{7} = 0
\]

Solving for \( x \):

\[
x = 9 - \sqrt{7}
\]

Since this factor is raised to the 4th power, the multiplicity of the zero \( x = 9 - \sqrt{7} \) is 4.

### Step 3: Summary of Zeros and Their Multiplicities

- \( x = -6 \) with multiplicity 2.
- \( x = 1 \) with multiplicity 2.
- \( x = 9 - \sqrt{7} \) with multiplicity 4.

### Final Answer:
The zeros of the function are:
- \( x = -6 \) (multiplicity 2),
- \( x = 1 \) (multiplicity 2),
- \( x = 9 - \sqrt{7} \) (multiplicity 4).

Do you have any questions or want further clarification?

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Here are 5 related questions for further exploration:
1. How do you factor a general quadratic equation like \( ax^2 + bx + c = 0 \)?
2. What does the multiplicity of a zero tell us about the graph of the function near that zero?
3. How do higher powers (like the fourth power) affect the behavior of a function near its zeros?
4. Can a quadratic equation have complex roots, and how would that affect the solution?
5. What are the methods to solve a quadratic equation if factoring is not possible?

**Tip**: The multiplicity of a root affects the flatness of the curve at that root. Higher multiplicities cause the graph to "flatten out" more at that zero.

Find all zeros of the function f(x) = (x^2 + 5x - 6)^2 (x - 9 + √7)^4 and their multiplicities.

Discover how to find all zeros and their multiplicities for the function f(x) = (x^2 + 5x - 6)^2 (x - 9 + √7)^4. This step-by-step solution covers key algebraic concepts such as quadratic equations, factoring, and polynomial functions, suitable for high school students (Grades 9-12).

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