At Westonci.ca, we connect you with the answers you need, thanks to our active and informed community. Experience the ease of finding precise answers to your questions from a knowledgeable community of experts. Get immediate and reliable solutions to your questions from a community of experienced professionals on our platform.
Sagot :
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]
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]
Thanks for using our service. We aim to provide the most accurate answers for all your queries. Visit us again for more insights. We appreciate your time. Please revisit us for more reliable answers to any questions you may have. We're glad you visited Westonci.ca. Return anytime for updated answers from our knowledgeable team.