To find [tex]\( P(0) \)[/tex], [tex]\( P(1) \)[/tex], and [tex]\( P(2) \)[/tex] for the polynomial [tex]\( p(t) = 2 + t + 2t^2 + t^3 \)[/tex], we need to substitute the values [tex]\( t = 0 \)[/tex], [tex]\( t = 1 \)[/tex], and [tex]\( t = 2 \)[/tex] into the polynomial and simplify.
Step-by-Step Solution:
1. Calculate [tex]\( P(0) \)[/tex]:
Substitute [tex]\( t = 0 \)[/tex] into the polynomial:
[tex]\[ p(0) = 2 + 0 + 2 \cdot 0^2 + 0^3 \][/tex]
Simplify:
[tex]\[ p(0) = 2 + 0 + 0 + 0 \][/tex]
[tex]\[ p(0) = 2 \][/tex]
So, [tex]\( P(0) = 2 \)[/tex].
2. Calculate [tex]\( P(1) \)[/tex]:
Substitute [tex]\( t = 1 \)[/tex] into the polynomial:
[tex]\[ p(1) = 2 + 1 + 2 \cdot 1^2 + 1^3 \][/tex]
Simplify:
[tex]\[ p(1) = 2 + 1 + 2 \cdot 1 + 1 \][/tex]
[tex]\[ p(1) = 2 + 1 + 2 + 1 \][/tex]
[tex]\[ p(1) = 6 \][/tex]
So, [tex]\( P(1) = 6 \)[/tex].
3. Calculate [tex]\( P(2) \)[/tex]:
Substitute [tex]\( t = 2 \)[/tex] into the polynomial:
[tex]\[ p(2) = 2 + 2 + 2 \cdot 2^2 + 2^3 \][/tex]
Simplify:
[tex]\[ p(2) = 2 + 2 + 2 \cdot 4 + 8 \][/tex]
[tex]\[ p(2) = 2 + 2 + 8 + 8 \][/tex]
[tex]\[ p(2) = 20 \][/tex]
So, [tex]\( P(2) = 20 \)[/tex].
Final Answer:
The values are:
[tex]\[ P(0) = 2 \][/tex]
[tex]\[ P(1) = 6 \][/tex]
[tex]\[ P(2) = 20 \][/tex]
