To find the value of the polynomial [tex]\(2x^3 - 3x^2 + 7\)[/tex] at [tex]\(x = 2\)[/tex] and [tex]\(x = -1\)[/tex], we need to substitute these values of [tex]\(x\)[/tex] into the polynomial and simplify. Let's go through each step-by-step.
### Evaluating the polynomial at [tex]\(x = 2\)[/tex]
1. Step 1: Substitute [tex]\(x = 2\)[/tex] into the polynomial
[tex]\[
2(2)^3 - 3(2)^2 + 7
\][/tex]
2. Step 2: Calculate the cubed term [tex]\(2^3\)[/tex]
[tex]\[
2^3 = 8 \implies 2 \cdot 8 = 16
\][/tex]
3. Step 3: Calculate the squared term [tex]\(2^2\)[/tex]
[tex]\[
2^2 = 4 \implies 3 \cdot 4 = 12
\][/tex]
4. Step 4: Combine the results
[tex]\[
16 - 12 + 7 = 11
\][/tex]
So, the value of the polynomial at [tex]\(x = 2\)[/tex] is [tex]\(11\)[/tex].
### Evaluating the polynomial at [tex]\(x = -1\)[/tex]
1. Step 1: Substitute [tex]\(x = -1\)[/tex] into the polynomial
[tex]\[
2(-1)^3 - 3(-1)^2 + 7
\][/tex]
2. Step 2: Calculate the cubed term [tex]\((-1)^3\)[/tex]
[tex]\[
(-1)^3 = -1 \implies 2 \cdot (-1) = -2
\][/tex]
3. Step 3: Calculate the squared term [tex]\((-1)^2\)[/tex]
[tex]\[
(-1)^2 = 1 \implies 3 \cdot 1 = 3
\][/tex]
4. Step 4: Combine the results
[tex]\[
-2 - 3 + 7 = 2
\][/tex]
So, the value of the polynomial at [tex]\(x = -1\)[/tex] is [tex]\(2\)[/tex].
### Summary of Results
- The value of the polynomial [tex]\(2x^3 - 3x^2 + 7\)[/tex] at [tex]\(x = 2\)[/tex] is [tex]\(11\)[/tex].
- The value of the polynomial [tex]\(2x^3 - 3x^2 + 7\)[/tex] at [tex]\(x = -1\)[/tex] is [tex]\(2\)[/tex].
Thus, the values are:
[tex]\[
\boxed{(11, 2)}
\][/tex]
