To find the value of [tex]\( b \)[/tex] in the polynomial function [tex]\( f(x) = ax^3 + bx^2 + cx + d \)[/tex] given the information that [tex]\(-1\)[/tex] and [tex]\(1\)[/tex] are roots and [tex]\((0, 3)\)[/tex] is the y-intercept, follow these steps:
1. Substitute the roots into the equation: Since [tex]\(-1\)[/tex] and [tex]\(1\)[/tex] are roots of the polynomial, this means [tex]\( f(-1) = 0 \)[/tex] and [tex]\( f(1) = 0 \)[/tex].
For [tex]\( f(-1) = 0 \)[/tex]:
[tex]\[
a(-1)^3 + b(-1)^2 + c(-1) + d = 0
\][/tex]
Simplify:
[tex]\[
-a + b - c + d = 0
\][/tex]
For [tex]\( f(1) = 0 \)[/tex]:
[tex]\[
a(1)^3 + b(1)^2 + c(1) + d = 0
\][/tex]
Simplify:
[tex]\[
a + b + c + d = 0
\][/tex]
2. Use the y-intercept information: Given that [tex]\((0, 3)\)[/tex] is the y-intercept means [tex]\( f(0) = 3 \)[/tex].
[tex]\[
f(0) = d = 3
\][/tex]
3. Form a system of equations: We now have:
[tex]\[
-a + b - c + d = 0 \quad \text{(equation 1)}
\][/tex]
[tex]\[
a + b + c + d = 0 \quad \text{(equation 2)}
\][/tex]
[tex]\[
d = 3 \quad \text{(equation 3)}
\][/tex]
4. Substitute [tex]\( d = 3 \)[/tex] into the first two equations:
Substitute [tex]\( d = 3 \)[/tex] into equation 1:
[tex]\[
-a + b - c + 3 = 0
\implies -a + b - c = -3 \quad \text{(equation 4)}
\][/tex]
Substitute [tex]\( d = 3 \)[/tex] into equation 2:
[tex]\[
a + b + c + 3 = 0
\implies a + b + c = -3 \quad \text{(equation 5)}
\][/tex]
5. Add equations 4 and 5: By adding equations 4 and 5, we can eliminate [tex]\( a \)[/tex] and [tex]\( c \)[/tex]:
[tex]\[
(-a + b - c) + (a + b + c) = -3 + (-3)
\implies 2b = -6
\implies b = -3
\][/tex]
So, the value of [tex]\( b \)[/tex] is [tex]\( -3 \)[/tex].
Therefore, the answer is [tex]\( \boxed{-3} \)[/tex].
