To determine which of the given choices could NOT be a zero of the polynomial [tex]\(4x^3 + 4x^2 - 11x - 6\)[/tex], we need to understand which potential rational zeros are possible based on the Rational Zero Theorem. Here's how you solve this step-by-step:
1. Identify the polynomial's coefficients:
- The polynomial is [tex]\(4x^3 + 4x^2 - 11x - 6\)[/tex].
- Leading coefficient (the coefficient of the highest power of [tex]\(x\)[/tex]) is 4.
- Constant term (the term without any [tex]\(x\)[/tex]) is -6.
2. Factors of the constant term (-6):
- The factors of -6 are [tex]\(\pm 1, \pm 2, \pm 3, \pm 6\)[/tex].
3. Factors of the leading coefficient (4):
- The factors of 4 are [tex]\(\pm 1, \pm 2, \pm 4\)[/tex].
4. Possible rational zeros (p/q values):
- The possible rational zeros are the ratios of the factors of the constant term to the factors of the leading coefficient: [tex]\(\frac{p}{q}\)[/tex].
- This results in the possible rational zeros: [tex]\(\pm 1, \pm \frac{1}{2}, \pm \frac{1}{4}, \pm 2, \pm \frac{3}{2}, \pm 3, \pm 6\)[/tex].
Thus, the possible rational zeros are:
[tex]\[
\{-6, -3, -2, -1, -\frac{3}{2}, -\frac{1}{2}, -\frac{1}{4}, \frac{1}{4}, \frac{1}{2}, 1, \frac{3}{2}, 2, 3, 6\}
\][/tex]
5. Verify the provided choices:
- The provided choices are 1, 6, 2, 3, 4.
6. Evaluate the polynomial at these points to determine which could NOT be a zero:
- We have determined the possible rational zeros, so now we can directly identify which numbers from the list of possible rational zeros include our given choices and which do not.
Here are the results:
- Choices included in the set of possible rational zeros: 1, 6, 2, 3
- Choice NOT included in the set: 4
Therefore, the number which could NOT be a zero of the polynomial [tex]\(4x^3 + 4x^2 - 11x - 6\)[/tex] is:
[tex]\[
\boxed{4}
\][/tex]
