To determine the value of [tex]\(\frac{f(1) + f(-1)}{f(2)}\)[/tex] for the polynomial [tex]\(f(x) = 5x^4 - 3x^3 + 2x^2 - 1\)[/tex], we need to evaluate the polynomial at the points [tex]\(x = 1\)[/tex], [tex]\(x = -1\)[/tex], and [tex]\(x = 2\)[/tex]. Let's go through the steps one by one.
First, calculate the value of [tex]\(f(1)\)[/tex]:
[tex]\[ f(1) = 5(1)^4 - 3(1)^3 + 2(1)^2 - 1 \][/tex]
[tex]\[ f(1) = 5 \cdot 1 - 3 \cdot 1 + 2 \cdot 1 - 1 \][/tex]
[tex]\[ f(1) = 5 - 3 + 2 - 1 \][/tex]
[tex]\[ f(1) = 3 \][/tex]
Next, calculate the value of [tex]\(f(-1)\)[/tex]:
[tex]\[ f(-1) = 5(-1)^4 - 3(-1)^3 + 2(-1)^2 - 1 \][/tex]
[tex]\[ f(-1) = 5 \cdot 1 - 3 \cdot (-1) + 2 \cdot 1 - 1 \][/tex]
[tex]\[ f(-1) = 5 + 3 + 2 - 1 \][/tex]
[tex]\[ f(-1) = 9 \][/tex]
Now, calculate the value of [tex]\(f(2)\)[/tex]:
[tex]\[ f(2) = 5(2)^4 - 3(2)^3 + 2(2)^2 - 1 \][/tex]
[tex]\[ f(2) = 5 \cdot 16 - 3 \cdot 8 + 2 \cdot 4 - 1 \][/tex]
[tex]\[ f(2) = 80 - 24 + 8 - 1 \][/tex]
[tex]\[ f(2) = 63 \][/tex]
We now have the values:
[tex]\[ f(1) = 3 \][/tex]
[tex]\[ f(-1) = 9 \][/tex]
[tex]\[ f(2) = 63 \][/tex]
Finally, we compute [tex]\(\frac{f(1) + f(-1)}{f(2)}\)[/tex]:
[tex]\[ \frac{f(1) + f(-1)}{f(2)} = \frac{3 + 9}{63} \][/tex]
[tex]\[ \frac{12}{63} = \frac{4}{21} \][/tex]
Thus, the value of [tex]\(\frac{f(1) + f(-1)}{f(2)}\)[/tex] is [tex]\(\frac{4}{21}\)[/tex].
So the correct choice is:
(2) [tex]\(\frac{4}{21}\)[/tex]
