Answer :
To solve for [tex]\(f(x)\)[/tex] when [tex]\(x = -5\)[/tex] using the polynomial function [tex]\(f(x) = 5x^4 + 45x^3 + 104x^2 + 36x + 80\)[/tex], we can substitute [tex]\(-5\)[/tex] into the function and evaluate it step-by-step:
1. Substitute [tex]\(-5\)[/tex] into the function:
[tex]\[
f(-5) = 5(-5)^4 + 45(-5)^3 + 104(-5)^2 + 36(-5) + 80
\][/tex]
2. Calculate each term:
- First Term:
[tex]\[
5(-5)^4 = 5 \times 625 = 3125
\][/tex]
- Second Term:
[tex]\[
45(-5)^3 = 45 \times (-125) = -5625
\][/tex]
- Third Term:
[tex]\[
104(-5)^2 = 104 \times 25 = 2600
\][/tex]
- Fourth Term:
[tex]\[
36(-5) = 36 \times (-5) = -180
\][/tex]
- Fifth Term:
[tex]\[
80
\][/tex]
3. Add up all the terms:
[tex]\[
f(-5) = 3125 - 5625 + 2600 - 180 + 80
\][/tex]
4. Simplify the expression:
- Start by adding and subtracting in sequence:
[tex]\[
3125 - 5625 = -2500
\][/tex]
[tex]\[
-2500 + 2600 = 100
\][/tex]
[tex]\[
100 - 180 = -80
\][/tex]
[tex]\[
-80 + 80 = 0
\][/tex]
So, the value of [tex]\( f(-5) \)[/tex] is [tex]\( 0 \)[/tex].
1. Substitute [tex]\(-5\)[/tex] into the function:
[tex]\[
f(-5) = 5(-5)^4 + 45(-5)^3 + 104(-5)^2 + 36(-5) + 80
\][/tex]
2. Calculate each term:
- First Term:
[tex]\[
5(-5)^4 = 5 \times 625 = 3125
\][/tex]
- Second Term:
[tex]\[
45(-5)^3 = 45 \times (-125) = -5625
\][/tex]
- Third Term:
[tex]\[
104(-5)^2 = 104 \times 25 = 2600
\][/tex]
- Fourth Term:
[tex]\[
36(-5) = 36 \times (-5) = -180
\][/tex]
- Fifth Term:
[tex]\[
80
\][/tex]
3. Add up all the terms:
[tex]\[
f(-5) = 3125 - 5625 + 2600 - 180 + 80
\][/tex]
4. Simplify the expression:
- Start by adding and subtracting in sequence:
[tex]\[
3125 - 5625 = -2500
\][/tex]
[tex]\[
-2500 + 2600 = 100
\][/tex]
[tex]\[
100 - 180 = -80
\][/tex]
[tex]\[
-80 + 80 = 0
\][/tex]
So, the value of [tex]\( f(-5) \)[/tex] is [tex]\( 0 \)[/tex].
