To find the remainder when [tex]\( t^4 - 3t^3 + t + 5 \)[/tex] is divided by [tex]\( t - 1 \)[/tex], we can use the Remainder Theorem. The Remainder Theorem states that if a polynomial [tex]\( f(t) \)[/tex] is divided by [tex]\( t - a \)[/tex], the remainder is [tex]\( f(a) \)[/tex].
In this case, our polynomial is [tex]\( f(t) = t^4 - 3t^3 + t + 5 \)[/tex], and we are dividing by [tex]\( t - 1 \)[/tex], so [tex]\( a = 1 \)[/tex].
We need to evaluate [tex]\( f(1) \)[/tex]:
[tex]\[ f(1) = 1^4 - 3(1^3) + 1 + 5 \][/tex]
Calculate each term:
1. [tex]\( 1^4 = 1 \)[/tex]
2. [tex]\( -3(1^3) = -3 \)[/tex]
3. [tex]\( 1 = 1 \)[/tex]
4. [tex]\( 5 = 5 \)[/tex]
Adding these together:
[tex]\[ f(1) = 1 - 3 + 1 + 5 \][/tex]
Compute the sum step-by-step:
[tex]\[ 1 - 3 = -2 \][/tex]
[tex]\[ -2 + 1 = -1 \][/tex]
[tex]\[ -1 + 5 = 4 \][/tex]
Therefore, the remainder when [tex]\( t^4 - 3t^3 + t + 5 \)[/tex] is divided by [tex]\( t - 1 \)[/tex] is [tex]\(\boxed{4}\)[/tex].
