Find the zeros of the function f(x) = x^3 – 8x^2 + 9x + 18, and describe the behavior of the graph at each zero.

Steps to Solve

1. Set the polynomial equal to zero

To find the zeros of the polynomial function ( f(x) = x^3 - 8x^2 + 9x + 18 ), we need to solve the equation:

$$ x^3 - 8x^2 + 9x + 18 = 0 $$

2. Use the Rational Root Theorem

We can use the Rational Root Theorem to find possible rational roots of the polynomial. The potential rational roots are factors of the constant term (18) divided by factors of the leading coefficient (1). The factors of 18 are ( \pm 1, \pm 2, \pm 3, \pm 6, \pm 9, \pm 18 ).

3. Test the possible rational roots

We will test each possible root by substituting it into the polynomial until we find one that gives ( f(x) = 0 ).

• Testing ( x = 1 ):

$$ f(1) = 1^3 - 8(1^2) + 9(1) + 18 = 1 - 8 + 9 + 18 = 20 \quad \text{(not a root)} $$

• Testing ( x = 2 ):

$$ f(2) = 2^3 - 8(2^2) + 9(2) + 18 = 8 - 32 + 18 + 18 = 12 \quad \text{(not a root)} $$

• Testing ( x = 3 ):

$$ f(3) = 3^3 - 8(3^2) + 9(3) + 18 = 27 - 72 + 27 + 18 = 0 \quad \text{(this is a root)} $$

4. Factor the polynomial

Since ( x = 3 ) is a root, we can perform synthetic or polynomial long division to factor the polynomial by ( x - 3 ):

Dividing ( f(x) ) by ( x - 3 ) gives:

$$ f(x) = (x - 3)(x^2 - 5x - 6) $$

5. Find additional zeros

Now, we need to factor ( x^2 - 5x - 6 ):

We can factor this further:

$$ x^2 - 5x - 6 = (x - 6)(x + 1) $$

So, the complete factorization is:

$$ f(x) = (x - 3)(x - 6)(x + 1) $$

6. Identify the zeros

The zeros of ( f(x) ) are ( x = 3, x = 6, x = -1 ).

7. Describe the behavior at zeros

To describe the behavior of the graph at each zero, we note:

• At ( x = 3 ) and ( x = 6 ), the function crosses the x-axis (as these factors have a multiplicity of 1).
• At ( x = -1 ), the function also crosses the x-axis.