To solve for [tex]\( f(-3) \)[/tex] given the function [tex]\( f(x) = 2x^4 - x^3 - 18x^2 + 9x \)[/tex], we need to substitute [tex]\( x \)[/tex] with [tex]\(-3\)[/tex] and then evaluate the result. Here are the steps:
1. Substitute [tex]\( x = -3 \)[/tex] in the function [tex]\( f(x) \)[/tex]:
[tex]\[
f(-3) = 2(-3)^4 - (-3)^3 - 18(-3)^2 + 9(-3)
\][/tex]
2. Evaluate each term separately:
- Compute [tex]\((-3)^4\)[/tex]:
[tex]\[
(-3)^4 = 81
\][/tex]
- Compute [tex]\(2 \cdot 81\)[/tex]:
[tex]\[
2 \cdot 81 = 162
\][/tex]
- Compute [tex]\((-3)^3\)[/tex]:
[tex]\[
(-3)^3 = -27
\][/tex]
- Compute [tex]\(-(-27)\)[/tex]:
[tex]\[
-(-27) = 27
\][/tex]
- Compute [tex]\((-3)^2\)[/tex]:
[tex]\[
(-3)^2 = 9
\][/tex]
- Compute [tex]\(18 \cdot 9\)[/tex]:
[tex]\[
18 \cdot 9 = 162
\][/tex]
- Compute [tex]\(18(-3)^2\)[/tex]:
[tex]\[
18 \cdot 9 = 162
\][/tex]
- Compute [tex]\(9 \cdot (-3)\)[/tex]:
[tex]\[
9 \cdot (-3) = -27
\][/tex]
3. Substitute these values back into the equation:
[tex]\[
f(-3) = 162 + 27 - 162 - 27
\][/tex]
4. Simplify the expression:
[tex]\[
f(-3) = 162 + 27 - 162 - 27 = 0
\][/tex]
Therefore, the value of [tex]\( f(-3) \)[/tex] is [tex]\( 0 \)[/tex].
