Certainly! Given a cubic polynomial function [tex]\( f(x) = ax^3 + bx^2 + cx + d \)[/tex] and the conditions provided, we need to write down the corresponding linear equations for each condition.
### Condition a: [tex]\( f(0) = -5 \)[/tex]
To determine the linear equation for [tex]\( f(0) \)[/tex]:
- Substitute [tex]\( x = 0 \)[/tex] into the polynomial function [tex]\( f(x) \)[/tex].
[tex]\[ f(0) = a(0)^3 + b(0)^2 + c(0) + d = d \][/tex]
Since [tex]\( f(0) = -5 \)[/tex],
[tex]\[ d = -5 \][/tex]
### Condition b: [tex]\( f(-1) = -5 \)[/tex]
To determine the linear equation for [tex]\( f(-1) \)[/tex]:
- Substitute [tex]\( x = -1 \)[/tex] into the polynomial function [tex]\( f(x) \)[/tex].
[tex]\[ f(-1) = a(-1)^3 + b(-1)^2 + c(-1) + d \][/tex]
[tex]\[ f(-1) = -a + b - c + d \][/tex]
Since [tex]\( f(-1) = -5 \)[/tex],
[tex]\[ -a + b - c + d = -5 \][/tex]
### Condition c: [tex]\( f(2) = 13 \)[/tex]
To determine the linear equation for [tex]\( f(2) \)[/tex]:
- Substitute [tex]\( x = 2 \)[/tex] into the polynomial function [tex]\( f(x) \)[/tex].
[tex]\[ f(2) = a(2)^3 + b(2)^2 + c(2) + d \][/tex]
[tex]\[ f(2) = 8a + 4b + 2c + d \][/tex]
Since [tex]\( f(2) = 13 \)[/tex],
[tex]\[ 8a + 4b + 2c + d = 13 \][/tex]
### Summary of Linear Equations
The linear equations that correspond to the given conditions are:
1. [tex]\( d = -5 \)[/tex]
2. [tex]\( -a + b - c + d = -5 \)[/tex]
3. [tex]\( 8a + 4b + 2c + d = 13 \)[/tex]
These equations can be used to determine the values of the coefficients [tex]\( a \)[/tex], [tex]\( b \)[/tex], [tex]\( c \)[/tex], and [tex]\( d \)[/tex] if required.
