QuestionPart of the fifth degree polynomial shown cannot be seen because of an inkblot. It is known that all five roots of the poly nomial are integers. What is the highest power of x - l that divides the polynomial? |
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| A: (x-1)^1 | B: (x-1)^2 | C: (x-1)^3 | D: (x-1)^4 | E: (x-1)^5 |
LLM Response: The graph of the polynomial shows an x-intercept at x=1. **What does it mean for a polynomial to have an x-intercept at a particular value?** It means that the value of the polynomial is zero at that x-value. In other words, x=1 is a root of the polynomial. **The problem states that all roots are integers. What does this tell us about the factors of the polynomial?** Since x=1 is a root, (x-1) must be a factor of the polynomial. Since all roots are integers, we know all the factors will be of the form (x - a) where 'a' is an integer. **We need to find the highest power of (x-1) that divides the polynomial. How do we do that?** We can look at the behavior of the graph at x=1. The graph doesn't simply cross the x-axis at x=1, it touches the x-axis and then turns back up. This indicates that (x-1) is a factor multiple times (i.e., it has a multiplicity greater than 1). **Let's analyze the multiplicity:** * **If (x-1) had a multiplicity of 1,** the graph would cross the x-axis directly. * **The graph's behavior suggests a multiplicity of at least 2.** Since we don't have enough information to definitively say the multiplicity is higher than 2, we assume the lowest possible multiplicity that fits the graph. **Therefore, the highest power of (x - 1) that divides the polynomial is (x-1)^2.** **Answer: B2**
Final Prediction: Puzzle #115: Solve this question with explanation of the intermediate steps. Your response must end with one of the selected answer options from A1, B2, C3, D4, or E5. Part of the fifth degree polynomial shown cannot be seen because of an inkblot. It is known that all five roots of the poly nomial are integers. What is the highest power of x - l that divides the polynomial? Options: A1: (x-1)^1, B2: (x-1)^2, C3: (x-1)^3, D4: (x-1)^4, E5: (x-1)^5. Predicted Answer = B Correct Answer = D