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Let's solve the polynomial equation step by step. The given function is:

\[
f(x) = x^3 + 8x^2 - 9x - 72
\]

### Step 1: Factor the polynomial
We will first try to find the rational zeros using the Rational Root Theorem, which states that any rational zero of the polynomial must be a factor of the constant term \(-72\) divided by the leading coefficient (which is 1 in this case).

The possible rational zeros are the factors of \(-72\):
\[
\pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 8, \pm 9, \pm 12, \pm 18, \pm 24, \pm 36, \pm 72
\]

Let's test some of these values by substituting into \( f(x) \).

### Step 2: Use synthetic division to find zeros
Testing \( x = -8 \):
\[
f(-8) = (-8)^3 + 8(-8)^2 - 9(-8) - 72 = -512 + 512 + 72 - 72 = 0
\]
Thus, \( x = -8 \) is a zero.

### Step 3: Polynomial division
Now, divide \( f(x) \) by \( (x + 8) \) to find the quotient polynomial:

Using synthetic division, we divide:

\[
x^3 + 8x^2 - 9x - 72 \div (x + 8)
\]

The quotient is \( x^2 - 9 \), so:

\[
f(x) = (x + 8)(x^2 - 9)
\]

### Step 4: Factor the quadratic
We can factor \( x^2 - 9 \) as a difference of squares:
\[
x^2 - 9 = (x - 3)(x + 3)
\]

Thus, the fully factored form of the polynomial is:
\[
f(x) = (x + 8)(x - 3)(x + 3)
\]

### Step 5: Find the zeros and their multiplicities
The zeros of the polynomial are:
\[
x = -8, x = 3, x = -3
\]

Each zero has a multiplicity of 1 since each factor appears only once.

### Step 6: Behavior of the graph at each zero
- At \( x = -8 \), the graph crosses the x-axis since the multiplicity is odd.
- At \( x = 3 \), the graph also crosses the x-axis (odd multiplicity).
- At \( x = -3 \), the graph crosses the x-axis as well.

### Final Answer:
The zeros are \( x = -8, x = -3, x = 3 \).

Let me know if you want further details or have any questions!

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Here are 5 related questions:
1. What is the significance of the multiplicity of a zero in polynomial graphs?
2. How does the Rational Root Theorem help in solving polynomial equations?
3. Can synthetic division be used for polynomials of any degree?
4. How do we determine whether a graph crosses or touches the x-axis at a zero?
5. What is the difference between real and complex zeros?

**Tip:** Always check the possible rational roots first before trying to factor complex polynomials!

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 + 8x^2 - 9x - 72

Learn how to find the zeros of the cubic polynomial f(x) = x^3 + 8x^2 - 9x - 72 using factoring and the Rational Root Theorem. This step-by-step guide is suitable for high school students (Grades 9-12) and covers key algebraic concepts.

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