To determine the result of the polynomial [tex]\( g(n) \)[/tex] multiplied by [tex]\( h(n) \)[/tex], we need to perform the multiplication and then simplify the resulting polynomial.
Given:
[tex]\[ g(n) = n^3 + n^2 + 2n \][/tex]
[tex]\[ h(n) = n - 4 \][/tex]
We need to compute [tex]\( (g \cdot h)(n) \)[/tex].
Step 1: Write out the multiplication explicitly:
[tex]\[ g(n) \cdot h(n) = (n^3 + n^2 + 2n)(n - 4) \][/tex]
Step 2: Distribute each term in [tex]\( g(n) \)[/tex] across each term in [tex]\( h(n) \)[/tex]:
[tex]\[ (n^3 + n^2 + 2n)(n - 4) = n^3 \cdot n + n^3 \cdot (-4) + n^2 \cdot n + n^2 \cdot (-4) + 2n \cdot n + 2n \cdot (-4) \][/tex]
Step 3: Simplify each term:
[tex]\[ = n^4 - 4n^3 + n^3 - 4n^2 + 2n^2 - 8n \][/tex]
Step 4: Combine like terms:
[tex]\[ = n^4 + (-4n^3 + n^3) + (-4n^2 + 2n^2) - 8n \][/tex]
[tex]\[ = n^4 - 3n^3 - 2n^2 - 8n \][/tex]
Thus, the simplified result of [tex]\( g(n) \cdot h(n) \)[/tex] is:
[tex]\[ n^4 - 3n^3 - 2n^2 - 8n \][/tex]
So the correct answer is:
A) [tex]\( n^4 - 3n^3 - 2n^2 - 8n \)[/tex]
