Answer :
To determine which of the given polynomials is a factor of [tex]\(2 x^3 - x^2 - 21 x + 18\)[/tex], we can use the Factor Theorem. The Factor Theorem states that a polynomial [tex]\(f(x)\)[/tex] has a factor [tex]\((x-c)\)[/tex] if and only if [tex]\(f(c) = 0\)[/tex].
Let's check each option one by one.
1. For [tex]\(x - 2\)[/tex]:
Evaluate the polynomial at [tex]\(x = 2\)[/tex]:
[tex]\[ f(2) = 2(2)^3 - (2)^2 - 21(2) + 18 \][/tex]
[tex]\[ f(2) = 2(8) - 4 - 42 + 18 \][/tex]
[tex]\[ f(2) = 16 - 4 - 42 + 18 \][/tex]
[tex]\[ f(2) = 28 - 46 \][/tex]
[tex]\[ f(2) = -18 \][/tex]
Since [tex]\(f(2) \neq 0\)[/tex], [tex]\(x - 2\)[/tex] is not a factor.
2. For [tex]\(x - 3\)[/tex]:
Evaluate the polynomial at [tex]\(x = 3\)[/tex]:
[tex]\[ f(3) = 2(3)^3 - (3)^2 - 21(3) + 18 \][/tex]
[tex]\[ f(3) = 2(27) - 9 - 63 + 18 \][/tex]
[tex]\[ f(3) = 54 - 9 - 63 + 18 \][/tex]
[tex]\[ f(3) = 63 - 63 \][/tex]
[tex]\[ f(3) = 0 \][/tex]
Since [tex]\(f(3) = 0\)[/tex], [tex]\(x - 3\)[/tex] is a factor.
3. For [tex]\(x + 2\)[/tex]:
Evaluate the polynomial at [tex]\(x = -2\)[/tex]:
[tex]\[ f(-2) = 2(-2)^3 - (-2)^2 - 21(-2) + 18 \][/tex]
[tex]\[ f(-2) = 2(-8) - 4 + 42 + 18 \][/tex]
[tex]\[ f(-2) = -16 - 4 + 42 + 18 \][/tex]
[tex]\[ f(-2) = -20 + 60 \][/tex]
[tex]\[ f(-2) = 40 \][/tex]
Since [tex]\(f(-2) \neq 0\)[/tex], [tex]\(x + 2\)[/tex] is not a factor.
4. For [tex]\(x + 3\)[/tex]:
Evaluate the polynomial at [tex]\(x = -3\)[/tex]:
[tex]\[ f(-3) = 2(-3)^3 - (-3)^2 - 21(-3) + 18 \][/tex]
[tex]\[ f(-3) = 2(-27) - 9 + 63 + 18 \][/tex]
[tex]\[ f(-3) = -54 - 9 + 63 + 18 \][/tex]
[tex]\[ f(-3) = -63 + 81 \][/tex]
[tex]\[ f(-3) = 18 \][/tex]
Since [tex]\(f(-3) \neq 0\)[/tex], [tex]\(x + 3\)[/tex] is not a factor.
Thus, of the given options, the only polynomial that is a factor of [tex]\(2 x^3 - x^2 - 21 x + 18\)[/tex] is [tex]\( \mathbf{x - 3} \)[/tex].
Let's check each option one by one.
1. For [tex]\(x - 2\)[/tex]:
Evaluate the polynomial at [tex]\(x = 2\)[/tex]:
[tex]\[ f(2) = 2(2)^3 - (2)^2 - 21(2) + 18 \][/tex]
[tex]\[ f(2) = 2(8) - 4 - 42 + 18 \][/tex]
[tex]\[ f(2) = 16 - 4 - 42 + 18 \][/tex]
[tex]\[ f(2) = 28 - 46 \][/tex]
[tex]\[ f(2) = -18 \][/tex]
Since [tex]\(f(2) \neq 0\)[/tex], [tex]\(x - 2\)[/tex] is not a factor.
2. For [tex]\(x - 3\)[/tex]:
Evaluate the polynomial at [tex]\(x = 3\)[/tex]:
[tex]\[ f(3) = 2(3)^3 - (3)^2 - 21(3) + 18 \][/tex]
[tex]\[ f(3) = 2(27) - 9 - 63 + 18 \][/tex]
[tex]\[ f(3) = 54 - 9 - 63 + 18 \][/tex]
[tex]\[ f(3) = 63 - 63 \][/tex]
[tex]\[ f(3) = 0 \][/tex]
Since [tex]\(f(3) = 0\)[/tex], [tex]\(x - 3\)[/tex] is a factor.
3. For [tex]\(x + 2\)[/tex]:
Evaluate the polynomial at [tex]\(x = -2\)[/tex]:
[tex]\[ f(-2) = 2(-2)^3 - (-2)^2 - 21(-2) + 18 \][/tex]
[tex]\[ f(-2) = 2(-8) - 4 + 42 + 18 \][/tex]
[tex]\[ f(-2) = -16 - 4 + 42 + 18 \][/tex]
[tex]\[ f(-2) = -20 + 60 \][/tex]
[tex]\[ f(-2) = 40 \][/tex]
Since [tex]\(f(-2) \neq 0\)[/tex], [tex]\(x + 2\)[/tex] is not a factor.
4. For [tex]\(x + 3\)[/tex]:
Evaluate the polynomial at [tex]\(x = -3\)[/tex]:
[tex]\[ f(-3) = 2(-3)^3 - (-3)^2 - 21(-3) + 18 \][/tex]
[tex]\[ f(-3) = 2(-27) - 9 + 63 + 18 \][/tex]
[tex]\[ f(-3) = -54 - 9 + 63 + 18 \][/tex]
[tex]\[ f(-3) = -63 + 81 \][/tex]
[tex]\[ f(-3) = 18 \][/tex]
Since [tex]\(f(-3) \neq 0\)[/tex], [tex]\(x + 3\)[/tex] is not a factor.
Thus, of the given options, the only polynomial that is a factor of [tex]\(2 x^3 - x^2 - 21 x + 18\)[/tex] is [tex]\( \mathbf{x - 3} \)[/tex].
