To find the value of [tex]\( n \)[/tex] for the given functions [tex]\( r(x) \)[/tex] and [tex]\( s(x) \)[/tex], given that their product equals [tex]\( x^4 - 81 \)[/tex], we need to perform polynomial multiplication and then compare coefficients.
First, let's write down the given functions:
[tex]\[ r(x) = x - 3 \][/tex]
[tex]\[ s(x) = x^3 + n x^2 + 3n x + 27 \][/tex]
We are given that:
[tex]\[ r(x) \cdot s(x) = x^4 - 81 \][/tex]
Let's multiply [tex]\( r(x) \)[/tex] and [tex]\( s(x) \)[/tex]:
[tex]\[
(x - 3) \cdot (x^3 + n x^2 + 3n x + 27)
\][/tex]
To expand this, we use the distributive property:
[tex]\[
= x \cdot (x^3 + n x^2 + 3n x + 27) - 3 \cdot (x^3 + n x^2 + 3n x + 27)
\][/tex]
[tex]\[
= x^4 + n x^3 + 3n x^2 + 27x - 3x^3 - 3n x^2 - 9n x - 81
\][/tex]
Now combine like terms:
[tex]\[
= x^4 + (n - 3)x^3 + (3n - 3n)x^2 + (27 - 9n)x - 81
\][/tex]
Simplify the expression:
[tex]\[
= x^4 + (n - 3)x^3 + 27x - 9n x - 81
\][/tex]
Given that [tex]\( r(x) \cdot s(x) = x^4 - 81 \)[/tex], we can compare coefficients from both sides of the equation.
Let's compare the coefficients of each term:
1. For the [tex]\( x^3 \)[/tex] term, the coefficient must be zero:
[tex]\[
n - 3 = 0
\][/tex]
Solving for [tex]\( n \)[/tex]:
[tex]\[
n = 3
\][/tex]
2. For the [tex]\( x \)[/tex] term, the coefficient must be zero:
[tex]\[
27 - 9n = 0
\][/tex]
Substitute [tex]\( n = 3 \)[/tex] into the equation:
[tex]\[
27 - 9 \cdot 3 = 0
\][/tex]
[tex]\[
27 - 27 = 0
\][/tex]
This condition is satisfied.
Therefore, the value of [tex]\( n \)[/tex] is [tex]\( \boxed{3} \)[/tex].
