To find the correct polynomial for which [tex]\(\left\{-\frac{1}{3}, 4\right\}\)[/tex] is the solution set, we first consider the given solutions and the fact that a quadratic polynomial can be derived using its roots.
Given the solutions [tex]\(\left\{-\frac{1}{3}, 4\right\}\)[/tex], we know that the polynomial can be expressed in its factored form as:
[tex]\[ (x + \frac{1}{3})(x - 4) = 0 \][/tex]
Let's express the polynomial in standard form by expanding this product:
Step 1: Distribute the terms inside the parentheses:
[tex]\[ \left( x + \frac{1}{3} \right) \left( x - 4 \right) = x(x - 4) + \frac{1}{3}(x - 4) \][/tex]
Step 2: Multiply each term:
[tex]\[ x^2 - 4x + \frac{1}{3} x - \frac{4}{3} \][/tex]
Step 3: Combine like terms:
[tex]\[ x^2 - \left( 4 - \frac{1}{3} \right) x - \frac{4}{3} \][/tex]
[tex]\[ x^2 - \left( \frac{12}{3} - \frac{1}{3} \right) x - \frac{4}{3} \][/tex]
[tex]\[ x^2 - \frac{11}{3} x - \frac{4}{3} \][/tex]
Step 4: Multiply the entire polynomial by the least common multiple of the denominators (in this case, 3) to clear the fractions:
[tex]\[ 3 \left( x^2 - \frac{11}{3} x - \frac{4}{3} \right) = 3x^2 - 11x - 4 \][/tex]
Comparing this polynomial with the given options, the correct polynomial that matches is:
[tex]\[ 3 x^{\wedge} 2-11 x+4=0 \][/tex]
Thus, the correct answer from the drop-down menu is:
[tex]\[ 3 x^{\wedge} 2-11 x+4=0 \][/tex]
