To find [tex]\( P(3) \)[/tex] for the polynomial [tex]\( P(t) = t^4 - t^3 + 3t^2 - 5 \)[/tex], follow these steps:
1. Substitute [tex]\( t = 3 \)[/tex] into the polynomial.
2. Compute each term separately.
Let's start by substituting [tex]\( t = 3 \)[/tex] into each term of the polynomial:
[tex]\[
P(t) = t^4 - t^3 + 3t^2 - 5
\][/tex]
Substituting [tex]\( t = 3 \)[/tex]:
[tex]\[
P(3) = 3^4 - 3^3 + 3 \cdot 3^2 - 5
\][/tex]
Now, calculate each term:
- First term: [tex]\( 3^4 \)[/tex]
[tex]\[
3^4 = 3 \times 3 \times 3 \times 3 = 81
\][/tex]
- Second term: [tex]\( -3^3 \)[/tex]
[tex]\[
3^3 = 3 \times 3 \times 3 = 27
\][/tex]
Therefore,
[tex]\[
-3^3 = -27
\][/tex]
- Third term: [tex]\( 3 \cdot 3^2 \)[/tex]
[tex]\[
3^2 = 3 \times 3 = 9
\][/tex]
Therefore,
[tex]\[
3 \cdot 9 = 27
\][/tex]
- Fourth term: [tex]\( -5 \)[/tex]
This term is already simplified as [tex]\( -5 \)[/tex].
Now, combine all the calculated terms together:
[tex]\[
P(3) = 81 - 27 + 27 - 5
\][/tex]
Perform the arithmetic operations step-by-step:
[tex]\[
81 - 27 = 54
\][/tex]
[tex]\[
54 + 27 = 81
\][/tex]
[tex]\[
81 - 5 = 76
\][/tex]
Thus,
[tex]\[
P(3) = 76
\][/tex]
Therefore, [tex]\( P(3) \)[/tex] is [tex]\( \boxed{76} \)[/tex].
