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!

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!