If [tex]\(-\frac{3}{2}\)[/tex] is a zero of the polynomial [tex]\(a x^3 - x^2 + x + 4\)[/tex], it means that substituting [tex]\(x = -\frac{3}{2}\)[/tex] into the polynomial should result in zero. This can be expressed mathematically as:
[tex]\[a \left(-\frac{3}{2}\right)^3 - \left(-\frac{3}{2}\right)^2 + \left(-\frac{3}{2}\right) + 4 = 0\][/tex]
Let's substitute [tex]\(x = -\frac{3}{2}\)[/tex] step-by-step:
1. Calculate [tex]\(\left(-\frac{3}{2}\right)^3\)[/tex]:
[tex]\[
\left(-\frac{3}{2}\right)^3 = -\frac{27}{8}
\][/tex]
2. Calculate [tex]\(\left(-\frac{3}{2}\right)^2\)[/tex]:
[tex]\[
\left(-\frac{3}{2}\right)^2 = \frac{9}{4}
\][/tex]
3. Substitute these into the polynomial and simplify:
[tex]\[
a \left(-\frac{27}{8}\right) - \frac{9}{4} - \frac{3}{2} + 4 = 0
\][/tex]
4. Combine like terms:
[tex]\[
-\frac{27a}{8} - \frac{9}{4} - \frac{3}{2} + 4 = 0
\][/tex]
To simplify the equation, let's find a common denominator for the fractions:
[tex]\[
-\frac{27a}{8} - \frac{18}{8} - \frac{12}{8} + \frac{32}{8} = 0
\][/tex]
Combining the fractions:
[tex]\[
-\frac{27a}{8} - \frac{18 + 12 - 32}{8} = 0
\][/tex]
Simplify inside the parentheses:
[tex]\[
-\frac{27a}{8} + \frac{2}{8} = 0
\][/tex]
Simplify further:
[tex]\[
-\frac{27a + 2}{8} = 0
\][/tex]
Since the fraction equals zero, the numerator must be zero:
[tex]\[
-27a + 2 = 0
\][/tex]
Solving for [tex]\(a\)[/tex]:
[tex]\[
-27a = -2
\][/tex]
[tex]\[
a = \frac{2}{27}
\][/tex]
Therefore, the value of [tex]\(a\)[/tex] is [tex]\(\boxed{\frac{2}{27}}\)[/tex].
