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Part 1

Find the zeros for the polynomial function and give the multiplicity for each zero.  State whether the graph crosses the​ x-axis or touches the​ x-axis and turns around at each zero.

​f(x)equals x cubed plus 7 x squared minus 4 x minus 28

We are tasked with finding the zeros of the polynomial function and determining their multiplicities, as well as describing whether the graph crosses the x-axis or touches the x-axis and turns around at each zero.

### Given Polynomial Function:
\[ f(x) = x^3 + 7x^2 - 4x - 28 \]

### Step 1: Factor the Polynomial
We will attempt to factor this polynomial to find its zeros.

#### Factor by grouping:
Group the terms:
\[
f(x) = (x^3 + 7x^2) - (4x + 28)
\]
Now, factor out the greatest common factor from each group:
\[
f(x) = x^2(x + 7) - 4(x + 7)
\]
Next, factor out the common binomial factor \((x + 7)\):
\[
f(x) = (x + 7)(x^2 - 4)
\]

#### Step 2: Factor the quadratic \((x^2 - 4)\)
Recognize that \(x^2 - 4\) is a difference of squares:
\[
x^2 - 4 = (x - 2)(x + 2)
\]

Thus, the fully factored form of \(f(x)\) is:
\[
f(x) = (x + 7)(x - 2)(x + 2)
\]

### Step 3: Find the Zeros
Now that we have factored the polynomial, we can find the zeros by setting each factor equal to zero:
\[
x + 7 = 0 \quad \Rightarrow \quad x = -7
\]
\[
x - 2 = 0 \quad \Rightarrow \quad x = 2
\]
\[
x + 2 = 0 \quad \Rightarrow \quad x = -2
\]

The zeros of the function are:
\[
x = -7, \quad x = 2, \quad x = -2
\]

### Step 4: Determine the Multiplicities
Since each factor appears only once in the factored form, each zero has multiplicity 1.

- Zero at \(x = -7\), multiplicity 1
- Zero at \(x = 2\), multiplicity 1
- Zero at \(x = -2\), multiplicity 1

### Step 5: Behavior at Each Zero
- If the multiplicity of a zero is odd, the graph crosses the x-axis at that point.
- If the multiplicity is even, the graph touches the x-axis and turns around at that point.

Since all zeros have multiplicity 1 (odd), the graph **crosses the x-axis** at each zero.

### Conclusion:
- The zeros are \(x = -7\), \(x = 2\), and \(x = -2\).
- Each zero has multiplicity 1.
- The graph crosses the x-axis at each of these zeros.

Would you like more details or explanations on any part of this solution?

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Here are some related questions to consider:

1. How do you factor polynomials by grouping?
2. What is the significance of multiplicity in polynomial functions?
3. How can you determine if a graph crosses or touches the x-axis by looking at the equation?
4. What are some examples of polynomials where the graph touches the x-axis but doesn't cross it?
5. How does the behavior of the graph change when the multiplicity of a zero is higher?

**Tip:** Always check if the polynomial can be factored by recognizing common patterns like the difference of squares or factoring by grouping.

Find the zeros for the polynomial function and give the multiplicity for each zero. State whether the graph crosses the x-axis or touches the x-axis and turns around at each zero. f(x) = x^3 + 7x^2 - 4x - 28

Learn how to find the zeros of the polynomial function f(x) = x^3 + 7x^2 - 4x - 28 and determine their multiplicities. This step-by-step solution explains polynomial factoring, the behavior of the graph at each zero, and the importance of multiplicity. Suitable for students in Grades 9-12.

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