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: To solve this problem, we need to identify the integer roots of the polynomial and determine the highest power of x - l that divides it. We can rewrite the given expression as: p(x) = x^5 - 11x^4 + 7 Since all five roots are integers, we know that there must be at least one root equal to 1. Let's factor out (x-1)^n from p(x), where n is the highest power of x - 1 that divides p(x): p(x) = (x-1)^n * q(x) where q(x) is a polynomial with integer coefficients. Let's find the remainder when p(x) is divided by x-1: p(1) = 1^5 - 11*1^4 + 7 = 0 This means that x-1 is a factor of p(x). Therefore, the highest power of x - 1 that divides p(x) is at least 1. Now, let's check if x-1 is a factor of p(x) more than once: p(x) = (x-1)^2 * r(x) where r(x) is a polynomial with integer coefficients. r(x) = (x-1) * s(x) Let's find the remainder when p(x) is divided by (x-1)^2: p(1) = 1^2 - 11*1^2 + 7 = 0 This means that (x-1)^2 is also a factor of p(x). Therefore, the highest power of x - 1 that divides p(x) is 2. So, the highest power of x - 1 that divides p(x) is B2: (x-1)^2.
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