Question

use the following function to answer parts a through c.
$f(x) = x^3 + 4x^2 - 218x - 221$

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. )
$-221, -17, -13, -1, 1, 13, 17, 221$

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. )
$-1$

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. )
$-1, -16.441, 13.441$

Explanation:

Step1: Recall Rational Zero Theorem

For a polynomial \( f(x) = a_nx^n + a_{n - 1}x^{n-1}+\dots+a_1x + a_0 \), the possible rational zeros are \( \pm\frac{\text{factors of }a_0}{\text{factors of }a_n} \). Here, \( f(x)=x^3 + 4x^2-218x - 221 \), so \( a_n = 1 \) (leading coefficient) and \( a_0=-221 \). Factors of \( 221 \) are \( 1,13,17,221 \), so possible rational zeros are \( \pm1,\pm13,\pm17,\pm221 \), which matches the given list.

Step2: Test \( x = - 1 \) via Synthetic Division

Set up synthetic division with root \( -1 \):

$$
LATEXBLOCK0
$$

Since the remainder is \( 0 \), \( x=-1 \) is a zero.

Step3: Factor the Polynomial

After synthetic division, the polynomial factors as \( (x + 1)(x^2+3x - 221) \). Now solve \( x^2+3x - 221 = 0 \) using quadratic formula \( x=\frac{-b\pm\sqrt{b^2 - 4ac}}{2a} \), where \( a = 1 \), \( b = 3 \), \( c=-221 \).

$$ x=\frac{-3\pm\sqrt{9+884}}{2}=\frac{-3\pm\sqrt{893}}{2}\approx\frac{-3\pm29.883}{2} $$

Calculating the two roots: \( \frac{-3 + 29.883}{2}\approx13.441 \), \( \frac{-3-29.883}{2}\approx - 16.441 \). So the zeros are \( -1,-16.441,13.441 \).

Answer:

-1, -16.441, 13.441