Answer :
To determine which of the given expressions is a factor of the polynomial [tex]\(3x^3 - 4x^2 - 13x - 6\)[/tex], we can utilize the Factor Theorem. The Factor Theorem states that a polynomial [tex]\(f(x)\)[/tex] has a factor [tex]\((x - k)\)[/tex] if and only if [tex]\(f(k) = 0\)[/tex].
We need to test each candidate to see if substituting the corresponding value into the polynomial yields zero. Let's denote the polynomial by [tex]\(p(x) = 3x^3 - 4x^2 - 13x - 6\)[/tex]. Now we will evaluate [tex]\(p(x)\)[/tex] for each root:
### Option A: [tex]\(x - 4\)[/tex]
To test if [tex]\(x - 4\)[/tex] is a factor, we substitute [tex]\(x = 4\)[/tex] into the polynomial:
[tex]\[ p(4) = 3(4)^3 - 4(4)^2 - 13(4) - 6 \][/tex]
[tex]\[ = 3(64) - 4(16) - 52 - 6 \][/tex]
[tex]\[ = 192 - 64 - 52 - 6 \][/tex]
[tex]\[ = 70 \][/tex]
Since [tex]\(p(4) = 70\)[/tex], [tex]\(x - 4\)[/tex] is not a factor.
### Option B: [tex]\(x + 3\)[/tex]
To test if [tex]\(x + 3\)[/tex] is a factor, we substitute [tex]\(x = -3\)[/tex] into the polynomial:
[tex]\[ p(-3) = 3(-3)^3 - 4(-3)^2 - 13(-3) - 6 \][/tex]
[tex]\[ = 3(-27) - 4(9) + 39 - 6 \][/tex]
[tex]\[ = -81 - 36 + 39 - 6 \][/tex]
[tex]\[ = -84 \][/tex]
Since [tex]\(p(-3) = -84\)[/tex], [tex]\(x + 3\)[/tex] is not a factor.
### Option C: [tex]\(x + 1\)[/tex]
To test if [tex]\(x + 1\)[/tex] is a factor, we substitute [tex]\(x = -1\)[/tex] into the polynomial:
[tex]\[ p(-1) = 3(-1)^3 - 4(-1)^2 - 13(-1) - 6 \][/tex]
[tex]\[ = 3(-1) - 4(1) + 13 - 6 \][/tex]
[tex]\[ = -3 - 4 + 13 - 6 \][/tex]
[tex]\[ = 0 \][/tex]
Since [tex]\(p(-1) = 0\)[/tex], [tex]\(x + 1\)[/tex] is a factor.
### Option D: [tex]\(x + 6\)[/tex]
To test if [tex]\(x + 6\)[/tex] is a factor, we substitute [tex]\(x = -6\)[/tex] into the polynomial:
[tex]\[ p(-6) = 3(-6)^3 - 4(-6)^2 - 13(-6) - 6 \][/tex]
[tex]\[ = 3(-216) - 4(36) + 78 - 6 \][/tex]
[tex]\[ = -648 - 144 + 78 - 6 \][/tex]
[tex]\[ = -720 \][/tex]
Since [tex]\(p(-6) = -720\)[/tex], [tex]\(x + 6\)[/tex] is not a factor.
Conclusively, the only factor is:
[tex]\[ \boxed{x + 1} \][/tex]
We need to test each candidate to see if substituting the corresponding value into the polynomial yields zero. Let's denote the polynomial by [tex]\(p(x) = 3x^3 - 4x^2 - 13x - 6\)[/tex]. Now we will evaluate [tex]\(p(x)\)[/tex] for each root:
### Option A: [tex]\(x - 4\)[/tex]
To test if [tex]\(x - 4\)[/tex] is a factor, we substitute [tex]\(x = 4\)[/tex] into the polynomial:
[tex]\[ p(4) = 3(4)^3 - 4(4)^2 - 13(4) - 6 \][/tex]
[tex]\[ = 3(64) - 4(16) - 52 - 6 \][/tex]
[tex]\[ = 192 - 64 - 52 - 6 \][/tex]
[tex]\[ = 70 \][/tex]
Since [tex]\(p(4) = 70\)[/tex], [tex]\(x - 4\)[/tex] is not a factor.
### Option B: [tex]\(x + 3\)[/tex]
To test if [tex]\(x + 3\)[/tex] is a factor, we substitute [tex]\(x = -3\)[/tex] into the polynomial:
[tex]\[ p(-3) = 3(-3)^3 - 4(-3)^2 - 13(-3) - 6 \][/tex]
[tex]\[ = 3(-27) - 4(9) + 39 - 6 \][/tex]
[tex]\[ = -81 - 36 + 39 - 6 \][/tex]
[tex]\[ = -84 \][/tex]
Since [tex]\(p(-3) = -84\)[/tex], [tex]\(x + 3\)[/tex] is not a factor.
### Option C: [tex]\(x + 1\)[/tex]
To test if [tex]\(x + 1\)[/tex] is a factor, we substitute [tex]\(x = -1\)[/tex] into the polynomial:
[tex]\[ p(-1) = 3(-1)^3 - 4(-1)^2 - 13(-1) - 6 \][/tex]
[tex]\[ = 3(-1) - 4(1) + 13 - 6 \][/tex]
[tex]\[ = -3 - 4 + 13 - 6 \][/tex]
[tex]\[ = 0 \][/tex]
Since [tex]\(p(-1) = 0\)[/tex], [tex]\(x + 1\)[/tex] is a factor.
### Option D: [tex]\(x + 6\)[/tex]
To test if [tex]\(x + 6\)[/tex] is a factor, we substitute [tex]\(x = -6\)[/tex] into the polynomial:
[tex]\[ p(-6) = 3(-6)^3 - 4(-6)^2 - 13(-6) - 6 \][/tex]
[tex]\[ = 3(-216) - 4(36) + 78 - 6 \][/tex]
[tex]\[ = -648 - 144 + 78 - 6 \][/tex]
[tex]\[ = -720 \][/tex]
Since [tex]\(p(-6) = -720\)[/tex], [tex]\(x + 6\)[/tex] is not a factor.
Conclusively, the only factor is:
[tex]\[ \boxed{x + 1} \][/tex]
