Answer :
To determine which polynomial functions have [tex]\((x+3)\)[/tex] as a factor, we should use the Factor Theorem. The Factor Theorem states that [tex]\((x-a)\)[/tex] is a factor of a polynomial [tex]\(f(x)\)[/tex] if and only if [tex]\(f(a) = 0\)[/tex]. Here, we need to check if [tex]\((x+3)\)[/tex] is a factor, so we substitute [tex]\(x = -3\)[/tex] into each polynomial and see if the result is zero.
Let's check each polynomial:
1. [tex]\(f(x) = x^4 - 12x^3 + 54x^2 - 108x + 81\)[/tex]:
Substitute [tex]\(x = -3\)[/tex]:
[tex]\[
f(-3) = (-3)^4 - 12(-3)^3 + 54(-3)^2 - 108(-3) + 81
\][/tex]
[tex]\[
= 81 - 12(-27) + 54(9) + 324 + 81
\][/tex]
[tex]\[
= 81 + 324 + 486 + 324 + 81
\][/tex]
[tex]\[
= 1296
\][/tex]
Since [tex]\(f(-3) \neq 0\)[/tex], [tex]\((x+3)\)[/tex] is not a factor.
2. [tex]\(f(x) = x^4 - 3x^3 - x + 3\)[/tex]:
Substitute [tex]\(x = -3\)[/tex]:
[tex]\[
f(-3) = (-3)^4 - 3(-3)^3 - (-3) + 3
\][/tex]
[tex]\[
= 81 + 81 + 3 + 3
\][/tex]
[tex]\[
= 168
\][/tex]
Since [tex]\(f(-3) \neq 0\)[/tex], [tex]\((x+3)\)[/tex] is not a factor.
3. [tex]\(f(x) = x^5 + 2x^4 - 23x^3 - 60x^2\)[/tex]:
Substitute [tex]\(x = -3\)[/tex]:
[tex]\[
f(-3) = (-3)^5 + 2(-3)^4 - 23(-3)^3 - 60(-3)^2
\][/tex]
[tex]\[
= -243 + 162 + 621 - 540
\][/tex]
[tex]\[
= 0
\][/tex]
Since [tex]\(f(-3) = 0\)[/tex], [tex]\((x+3)\)[/tex] is a factor.
4. [tex]\(f(x) = x^5 + 5x^4 - 3x^3 - 29x^2 + 2x + 24\)[/tex]:
Substitute [tex]\(x = -3\)[/tex]:
[tex]\[
f(-3) = (-3)^5 + 5(-3)^4 - 3(-3)^3 - 29(-3)^2 + 2(-3) + 24
\][/tex]
[tex]\[
= -243 + 405 + 81 - 261 - 6 + 24
\][/tex]
[tex]\[
= 0
\][/tex]
Since [tex]\(f(-3) = 0\)[/tex], [tex]\((x+3)\)[/tex] is a factor.
Thus, the polynomial functions for which [tex]\((x+3)\)[/tex] is a factor are:
- [tex]\(f(x) = x^5 + 2x^4 - 23x^3 - 60x^2\)[/tex]
- [tex]\(f(x) = x^5 + 5x^4 - 3x^3 - 29x^2 + 2x + 24\)[/tex]
Let's check each polynomial:
1. [tex]\(f(x) = x^4 - 12x^3 + 54x^2 - 108x + 81\)[/tex]:
Substitute [tex]\(x = -3\)[/tex]:
[tex]\[
f(-3) = (-3)^4 - 12(-3)^3 + 54(-3)^2 - 108(-3) + 81
\][/tex]
[tex]\[
= 81 - 12(-27) + 54(9) + 324 + 81
\][/tex]
[tex]\[
= 81 + 324 + 486 + 324 + 81
\][/tex]
[tex]\[
= 1296
\][/tex]
Since [tex]\(f(-3) \neq 0\)[/tex], [tex]\((x+3)\)[/tex] is not a factor.
2. [tex]\(f(x) = x^4 - 3x^3 - x + 3\)[/tex]:
Substitute [tex]\(x = -3\)[/tex]:
[tex]\[
f(-3) = (-3)^4 - 3(-3)^3 - (-3) + 3
\][/tex]
[tex]\[
= 81 + 81 + 3 + 3
\][/tex]
[tex]\[
= 168
\][/tex]
Since [tex]\(f(-3) \neq 0\)[/tex], [tex]\((x+3)\)[/tex] is not a factor.
3. [tex]\(f(x) = x^5 + 2x^4 - 23x^3 - 60x^2\)[/tex]:
Substitute [tex]\(x = -3\)[/tex]:
[tex]\[
f(-3) = (-3)^5 + 2(-3)^4 - 23(-3)^3 - 60(-3)^2
\][/tex]
[tex]\[
= -243 + 162 + 621 - 540
\][/tex]
[tex]\[
= 0
\][/tex]
Since [tex]\(f(-3) = 0\)[/tex], [tex]\((x+3)\)[/tex] is a factor.
4. [tex]\(f(x) = x^5 + 5x^4 - 3x^3 - 29x^2 + 2x + 24\)[/tex]:
Substitute [tex]\(x = -3\)[/tex]:
[tex]\[
f(-3) = (-3)^5 + 5(-3)^4 - 3(-3)^3 - 29(-3)^2 + 2(-3) + 24
\][/tex]
[tex]\[
= -243 + 405 + 81 - 261 - 6 + 24
\][/tex]
[tex]\[
= 0
\][/tex]
Since [tex]\(f(-3) = 0\)[/tex], [tex]\((x+3)\)[/tex] is a factor.
Thus, the polynomial functions for which [tex]\((x+3)\)[/tex] is a factor are:
- [tex]\(f(x) = x^5 + 2x^4 - 23x^3 - 60x^2\)[/tex]
- [tex]\(f(x) = x^5 + 5x^4 - 3x^3 - 29x^2 + 2x + 24\)[/tex]
