Question

the following function is given.
$f(x)=x^3 - 2x^2 - 9x + 18$

a. list all rational zeros that are possible according to the rational zero theorem.
(use a comma to separate answers as needed.)

b. use synthetic division to test several possible rational zeros in order to identify one actual zero.
one rational zero of the given function is
(simplify your answer.)

c. use the zero from part (b) to find all the zeros of the polynomial function.
the zeros of the function $f(x)=x^3 - 2x^2 - 9x + 18$ are
(simplify your answer. type an integer or a fraction. use a comma to separate answers as needed.)

Explanation:

Part (a)

Step 1: Identify leading coefficient and constant term

For the polynomial \( f(x) = x^3 - 2x^2 - 9x + 18 \), the leading coefficient (coefficient of \( x^3 \)) is \( 1 \), and the constant term is \( 18 \).

Step 2: Apply Rational Zero Theorem

The Rational Zero Theorem states that possible rational zeros are of the form \( \frac{p}{q} \), where \( p \) is a factor of the constant term and \( q \) is a factor of the leading coefficient.

• Factors of \( 18 \) ( \( p \) ): \( \pm 1, \pm 2, \pm 3, \pm 6, \pm 9, \pm 18 \)
• Factors of \( 1 \) ( \( q \) ): \( \pm 1 \)

So, possible rational zeros are \( \frac{\pm 1, \pm 2, \pm 3, \pm 6, \pm 9, \pm 18}{\pm 1} \), which simplifies to \( \pm 1, \pm 2, \pm 3, \pm 6, \pm 9, \pm 18 \).

Step 1: Test possible rational zeros

Let's test \( x = 2 \) using synthetic division. The coefficients of the polynomial are \( 1, -2, -9, 18 \).

Set up synthetic division for \( x = 2 \):

$$
LATEXBLOCK0
$$

The remainder is \( 0 \), so \( x = 2 \) is a root. We could also test other values, but \( x = 2 \) works.

Step 1: Use the zero from part (b) to factor the polynomial

From part (b), we found that \( x = 2 \) is a zero, so \( (x - 2) \) is a factor. The quotient from synthetic division (the coefficients \( 1, 0, -9 \)) gives the quadratic factor \( x^2 + 0x - 9 = x^2 - 9 \).

Step 2: Factor the quadratic

The quadratic \( x^2 - 9 \) is a difference of squares, so it factors as \( (x - 3)(x + 3) \).

Step 3: Find all zeros

Setting each factor equal to zero:

• \( x - 2 = 0 \) gives \( x = 2 \)
• \( x - 3 = 0 \) gives \( x = 3 \)
• \( x + 3 = 0 \) gives \( x = -3 \)

Answer:

\( \pm 1, \pm 2, \pm 3, \pm 6, \pm 9, \pm 18 \)

Part (b)