To find the polynomial given the solution set [tex]\(\left\{-\frac{1}{3}, 4\right\}\)[/tex], follow these detailed steps:
1. Identify the solutions and form factors:
- The solutions (or roots) given are [tex]\(-\frac{1}{3}\)[/tex] and [tex]\(4\)[/tex].
- Each root corresponds to a factor of the polynomial:
- For the root [tex]\(-\frac{1}{3}\)[/tex], the factor is [tex]\((x - (-\frac{1}{3}))\)[/tex] which simplifies to [tex]\((x + \frac{1}{3})\)[/tex].
- For the root [tex]\(4\)[/tex], the factor is [tex]\((x - 4)\)[/tex].
2. Construct the polynomial:
- The polynomial can be formed by multiplying these factors together:
[tex]\[
(x + \frac{1}{3})(x - 4)
\][/tex]
3. Expand the polynomial:
- To find the polynomial in standard form, expand the product:
[tex]\[
(x + \frac{1}{3})(x - 4) = x^2 - 4x + \frac{1}{3}x - \frac{4}{3}
\][/tex]
4. Combine like terms:
- Combine the [tex]\(x\)[/tex] terms:
[tex]\[
x^2 - 4x + \frac{1}{3}x - \frac{4}{3}
\][/tex]
[tex]\[
x^2 - \frac{11}{3}x - \frac{4}{3}
\][/tex]
So, the polynomial in standard form is:
[tex]\[
\boxed{x^2 - 3.66666666666667x - 1.33333333333333}
\][/tex]
Therefore, [tex]\(\left\{-\frac{1}{3}, 4\right\}\)[/tex] is the solution set of the polynomial [tex]\(x^2 - 3.66666666666667x - 1.33333333333333\)[/tex]. Select this polynomial from the drop-down menu to correctly answer the question.
