To find the correct polynomial from the given solution set [tex]\(\left\{-\frac{1}{3}, 4\right\}\)[/tex], let's go through the steps to obtain the polynomial.
1. Identify the roots: The polynomial has roots at [tex]\(x = -\frac{1}{3}\)[/tex] and [tex]\(x = 4\)[/tex].
2. Construct the factors: Each root corresponds to a factor of the polynomial:
[tex]\[
(x - \left(-\frac{1}{3}\right)) \quad \text{and} \quad (x - 4)
\][/tex]
Simplifying the first factor:
[tex]\[
x + \frac{1}{3}
\][/tex]
3. Form the polynomial: Multiply these factors together:
[tex]\[
(x + \frac{1}{3})(x - 4)
\][/tex]
4. Expand the expression: Now, distribute the terms:
[tex]\[
(x + \frac{1}{3})(x - 4) = x^2 - 4x + \frac{1}{3}x - \frac{4}{3}
\][/tex]
5. Combine like terms: Add the terms together in standard form:
[tex]\[
x^2 - \left(\frac{12}{3} - \frac{1}{3}\right)x - \frac{4}{3} = x^2 - \frac{11}{3}x - \frac{4}{3}
\][/tex]
6. Clear the fractions: Multiply the entire polynomial by 3 to eliminate the fractions:
[tex]\[
3 \cdot \left(x^2 - \frac{11}{3}x - \frac{4}{3}\right) = 3x^2 - 11x + 4
\][/tex]
So, the polynomial with solution set [tex]\(\left\{-\frac{1}{3}, 4\right\}\)[/tex] is:
[tex]\[
3x^2 - 11x + 4 = 0
\][/tex]
Therefore, the correct polynomial from the given options is:
[tex]\[
3x^2 - 11x + 4 = 0
\][/tex]
