Answer :

GinnyAnswer
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].

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