Evaluate the Function f(x)=2x^5-3x^4-9x^3+8x^2+2 at x=-3

$f (x) = 2 x^{5} - 3 x^{4} - 9 x^{3} + 8 x^{2} + 2$ at $x = - 3$

Replace the variable $x$ with $- 3$ in the expression.

$f (- 3) = 2 {(- 3)}^{5} - 3 {(- 3)}^{4} - 9 {(- 3)}^{3} + 8 {(- 3)}^{2} + 2$

Simplify the result.

Simplify each term.

Raise $- 3$ to the power of $5$ .

$f (- 3) = 2 \cdot - 243 - 3 {(- 3)}^{4} - 9 {(- 3)}^{3} + 8 {(- 3)}^{2} + 2$

Multiply $2$ by $- 243$ .

$f (- 3) = - 486 - 3 {(- 3)}^{4} - 9 {(- 3)}^{3} + 8 {(- 3)}^{2} + 2$

Multiply $- 3$ by ${(- 3)}^{4}$ by adding the exponents.

Multiply $- 3$ by ${(- 3)}^{4}$ .

Raise $- 3$ to the power of $1$ .

$f (- 3) = - 486 + (- 3) {(- 3)}^{4} - 9 {(- 3)}^{3} + 8 {(- 3)}^{2} + 2$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f (- 3) = - 486 + {(- 3)}^{1 + 4} - 9 {(- 3)}^{3} + 8 {(- 3)}^{2} + 2$

Add $1$ and $4$ .

$f (- 3) = - 486 + {(- 3)}^{5} - 9 {(- 3)}^{3} + 8 {(- 3)}^{2} + 2$

Raise $- 3$ to the power of $5$ .

$f (- 3) = - 486 - 243 - 9 {(- 3)}^{3} + 8 {(- 3)}^{2} + 2$

Raise $- 3$ to the power of $3$ .

$f (- 3) = - 486 - 243 - 9 \cdot - 27 + 8 {(- 3)}^{2} + 2$

Multiply $- 9$ by $- 27$ .

$f (- 3) = - 486 - 243 + 243 + 8 {(- 3)}^{2} + 2$

Raise $- 3$ to the power of $2$ .

$f (- 3) = - 486 - 243 + 243 + 8 \cdot 9 + 2$

Multiply $8$ by $9$ .

$f (- 3) = - 486 - 243 + 243 + 72 + 2$

Simplify by adding and subtracting.

Subtract $243$ from $- 486$ .

$f (- 3) = - 729 + 243 + 72 + 2$

Add $- 729$ and $243$ .

$f (- 3) = - 486 + 72 + 2$

Add $- 486$ and $72$ .

$f (- 3) = - 414 + 2$

Add $- 414$ and $2$ .

$f (- 3) = - 412$

The final answer is $- 412$ .

$- 412$