Let's evaluate the polynomial [tex]\( -7x^3 + 3x^2 + 11x - 9 \)[/tex] at [tex]\( x = -2 \)[/tex]:
1. Step 1: Substitute [tex]\( x = -2 \)[/tex] into the polynomial.
[tex]\[
-7(-2)^3 + 3(-2)^2 + 11(-2) - 9
\][/tex]
2. Step 2: Evaluate each term individually.
- Term 1: [tex]\( -7(-2)^3 \)[/tex]
[tex]\[
(-2)^3 = -8
\][/tex]
[tex]\[
-7 \times -8 = 56
\][/tex]
- Term 2: [tex]\( 3(-2)^2 \)[/tex]
[tex]\[
(-2)^2 = 4
\][/tex]
[tex]\[
3 \times 4 = 12
\][/tex]
- Term 3: [tex]\( 11(-2) \)[/tex]
[tex]\[
11 \times -2 = -22
\][/tex]
- Term 4: [tex]\( -9 \)[/tex] (Constant term remains the same.)
[tex]\[
-9
\][/tex]
3. Step 3: Add all the evaluated terms together.
[tex]\[
56 + 12 - 22 - 9
\][/tex]
4. Step 4: Combine the values step-by-step.
First, add the positive values:
[tex]\[
56 + 12 = 68
\][/tex]
Next, subtract the first negative value:
[tex]\[
68 - 22 = 46
\][/tex]
Finally, subtract the second negative value:
[tex]\[
46 - 9 = 37
\][/tex]
Therefore, the value of the polynomial [tex]\( -7x^3 + 3x^2 + 11x - 9 \)[/tex] when [tex]\( x = -2 \)[/tex] is [tex]\( \boxed{37} \)[/tex].
