To find a polynomial with the given roots [tex]\(\left\{-\frac{1}{3}, 4\right\}\)[/tex], follow these steps:
1. Recall that if [tex]\(r\)[/tex] is a root of a polynomial, then [tex]\((x - r)\)[/tex] is a factor of that polynomial. Therefore, for the roots [tex]\(-\frac{1}{3}\)[/tex] and [tex]\(4\)[/tex], the corresponding factors are [tex]\((x + \frac{1}{3})\)[/tex] and [tex]\((x - 4)\)[/tex].
2. To find the polynomial, we need to multiply these factors together:
[tex]\[
(x + \frac{1}{3})(x - 4)
\][/tex]
3. Distribute to multiply the two binomials:
[tex]\[
(x + \frac{1}{3})(x - 4) = x(x - 4) + \frac{1}{3}(x - 4)
\][/tex]
4. Expand each term:
[tex]\[
= x^2 - 4x + \frac{1}{3}x - \frac{4}{3}
\][/tex]
5. Combine like terms:
[tex]\[
= x^2 - \left(4 - \frac{1}{3}\right)x - \frac{4}{3}
\][/tex]
[tex]\[
= x^2 - \frac{12}{3}x + \frac{1}{3}x - \frac{4}{3}
\][/tex]
[tex]\[
= x^2 - \frac{12}{3}x + \frac{1}{3}x - \frac{4}{3}
\][/tex]
6. Simplify the coefficients:
[tex]\[
= x^2 - \frac{11}{3}x - \frac{4}{3}
\][/tex]
Thus, the polynomial in standard form with the given roots [tex]\(\left\{-\frac{1}{3}, 4\right\}\)[/tex] is
[tex]\[
x^2 - 3.66666666666667x - 1.33333333333333
\][/tex]
So, the correct answer is:
[tex]\[
\boxed{x^2 - 3.66666666666667x - 1.33333333333333}
\][/tex]
