Answered

Discover a world of knowledge and community-driven answers at IDNLearn.com today. Our platform provides trustworthy answers to help you make informed decisions quickly and easily.

Find [tex]\( P(0), P(1), P(2) \)[/tex] if [tex]\( P(t) = 2 + t + 2t^2 + t^3 \)[/tex].

Sagot :

GinnyAnsweridn
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]
We value your presence here. Keep sharing knowledge and helping others find the answers they need. This community is the perfect place to learn together. IDNLearn.com has the solutions to your questions. Thanks for stopping by, and come back for more insightful information.