To determine which expressions are factors of the polynomial [tex]\( f(x) = x^5 + x^4 - 27x^3 - x^2 + 146x - 120 \)[/tex], we consider the given values where [tex]\( f(x) \)[/tex] is zero. These are the points where the polynomial intersects the x-axis, effectively representing the roots of the polynomial.
The roots are:
- [tex]\( f(-5) = 0 \)[/tex]
- [tex]\( f(-3) = 0 \)[/tex]
- [tex]\( f(4) = 0 \)[/tex]
These roots correspond directly to the factors of the polynomial. For each root [tex]\( x = a \)[/tex], the factor is [tex]\( x - a \)[/tex].
1. For [tex]\( x = -5 \)[/tex]:
- The corresponding factor is [tex]\( x + 5 \)[/tex].
2. For [tex]\( x = -3 \)[/tex]:
- The corresponding factor is [tex]\( x + 3 \)[/tex].
3. For [tex]\( x = 4 \)[/tex]:
- The corresponding factor is [tex]\( x - 4 \)[/tex].
Thus, the expressions that are factors of the polynomial [tex]\( f(x) = x^5 + x^4 - 27x^3 - x^2 + 146x - 120 \)[/tex] are:
[tex]\[
x + 5, \quad x + 3, \quad x - 4
\][/tex]
Therefore, the correct answers are:
- [tex]\( x + 5 \)[/tex]
- [tex]\( x + 3 \)[/tex]
- [tex]\( x - 4 \)[/tex]
