Question

use the following function to answer parts a through c.
f(x) = x³ + 4x² - 218x - 221
a. list all rational zeros that are possible according to the rational zero theorem.
(type an integer or a simplified fraction. use a comma to separate answers as needed. type each answer only once.)
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 are
(simplify your answer. use a comma to separate answers as needed. type an integer or decimal rounded to three decimal places as needed.)

Explanation:

Step1: Apply Rational Zero Theorem

For the polynomial \( f(x) = x^3 + 4x^2 - 218x - 221 \), the constant term is \( -221 \) and the leading coefficient is \( 1 \). The factors of \( -221 \) are \( \pm1, \pm13, \pm17, \pm221 \), and the factors of \( 1 \) are \( \pm1 \). By the Rational Zero Theorem, possible rational zeros are \( \pm1, \pm13, \pm17, \pm221 \).

Step2: Test possible zeros with synthetic division

Test \( x = 13 \):

$$
LATEXBLOCK0
$$

Not a zero. Test \( x = 17 \):

$$
LATEXBLOCK1
$$

Not a zero. Test \( x = -1 \):

$$
LATEXBLOCK2
$$

So \( x = -1 \) is a zero.

Step3: Factor the polynomial

Since \( x = -1 \) is a zero, \( (x + 1) \) is a factor. The quotient from synthetic division is \( x^2 + 3x - 221 \). Now solve \( x^2 + 3x - 221 = 0 \) using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 1 \), \( b = 3 \), \( c = -221 \).

$$ x = \frac{-3 \pm \sqrt{9 + 884}}{2} = \frac{-3 \pm \sqrt{893}}{2} \approx \frac{-3 \pm 29.883}{2} $$

So the zeros are \( x = -1 \), \( x \approx \frac{-3 + 29.883}{2} \approx 13.441 \), \( x \approx \frac{-3 - 29.883}{2} \approx -16.441 \).

Answer:

Part a:

The possible rational zeros are \( -221, -17, -13, -1, 1, 13, 17, 221 \).

Part b:

One rational zero is \( -1 \).

Part c:

The zeros of the function are \( -1, -16.441, 13.441 \) (rounded to three decimal places).