To find the zeros of the function [tex]\( f(x) = 2x^3 + x^2 - 12x + 9 \)[/tex], we need to determine the values of [tex]\( x \)[/tex] for which [tex]\( f(x) = 0 \)[/tex].
Step-by-step solution:
1. Set the function equal to zero:
[tex]\[
2x^3 + x^2 - 12x + 9 = 0
\][/tex]
2. Identify possible rational roots using the Rational Root Theorem: The Rational Root Theorem states that any rational solution of the polynomial equation [tex]\( a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0 = 0 \)[/tex] will be a fraction [tex]\( \frac{p}{q} \)[/tex], where [tex]\( p \)[/tex] is a factor of the constant term [tex]\( a_0 \)[/tex] and [tex]\( q \)[/tex] is a factor of the leading coefficient [tex]\( a_n \)[/tex]. Here, [tex]\( a_0 = 9 \)[/tex] and [tex]\( a_n = 2 \)[/tex].
Hence possible rational roots to test are [tex]\( \pm 1, \pm 3, \pm 9, \pm \frac{1}{2}, \pm \frac{3}{2}, \pm \frac{9}{2} \)[/tex].
3. Test the possible rational roots:
After performing synthetic division or substituting into the polynomial, we find that the actual roots (zeros) of the polynomial are:
- [tex]\( x = -3 \)[/tex]
- [tex]\( x = 1 \)[/tex]
- [tex]\( x = \frac{3}{2} \)[/tex]
4. List the roots from smallest to largest:
The zeros of the function [tex]\( f(x) \)[/tex] are [tex]\( x = -3 \)[/tex], [tex]\( x = 1 \)[/tex], and [tex]\( x = \frac{3}{2} \)[/tex].
So, the final ordered list of zeros is:
[tex]\[
x = -3, 1, \frac{3}{2}
\][/tex]
