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: [INST] Solve this question with explanation of the intermediate steps. Your response m ust 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. [/INST] To solve this problem, we need to find the highest power of x - l that divides the polynomial. Since all five roots of the polynomial are integers, we can use the fact that the polynomial is divisible by (x - l)^5. Let's assume that the polynomial is of the form: (x - l)^5 + a(x - l)^4 + b(x - l)^3 + c(x - l)^2 + d(x - l) + e where a, b, c, d, and e are constants. Since all five roots are integers, we can write: (x - l)^5 + a(x - l)^4 + b(x - l)^3 + c(x - l)^2 + d(x - l) + e = 0 Now, let's consider the possible values of the highest power of x - l that divides the polynomial: A1: (x-1)^1 B2: (x-1)^2 C3: (x-1)^3 D4: (x-1)^4 E5: (x-1)^5 Let's test each of these options to see which one divides the polynomial. Option A1: (x-1)^1 If we substitute x = l into the polynomial, we get: (x - l)^5 + a(x - l)^4 + b(x - l)^3 + c(x - l)^2 + d(x - l) + e = 0 Substituting x = l into the polynomial gives: (0)^5 + a(0)^4 + b(0)^3 + c(0)^2 + d(0) + e = 0 This simplifies to: 0 + 0 + 0 + 0 + d + e = 0 Since d and e are constants, this equation is always true. Therefore, (x-1)^1 does not divide the polynomial. Option B2: (x-1)^2 If we substitute x = l into the polynomial, we get: (x - l)^5 + a(x - l)^4 + b(x - l)^3 + c(x - l)^2 + d(x - l) + e = 0 Substituting x = l into the polynomial gives: (0)^5 + a(0)^4 + b(0)^3 + c(0)^2 + d(0) + e = 0 This simplifies to: 0 + a + b + c + d + e = 0 Since d and e are constants, this equation is always true. Therefore, (x-1)^2 does not divide the polynomial. Option C3: (x-1)^3 If we substitute x = l into the polynomial, we get: (x - l)^5 + a(x - l)^4 + b(x - l)^3 + c(x - l)^2 + d(x - l) + e = 0 Substituting x = l into the polynomial gives: (0)^5 + a(0)^4 + b(0)^3 + c(0)^2 + d(0) + e = 0 This simplifies to: 0 + a + b + c + d + e = 0 Since d and e are constants, this equation is always true. Therefore, (x-1)^3 does not divide the polynomial. Option D4: (x-1)^4 If we substitute x = l into the polynomial, we get: (x - l)^5 + a(x - l)^4 + b(x - l)^3 + c(x - l)^2 + d(x - l) + e = 0 Substituting x = l into the polynomial gives: (0)^5 + a(0)^4 + b(0)^3 + c(0)^2 + d(0) + e = 0 This simplifies to: 0 + a + b + c + d + e = 0 Since d and e are constants, this equation is always true. Therefore, (x-1)^4 does not divide the polynomial. Option E5: (x-1)^5 If we substitute x = l into the polynomial, we get: (x - l)^5 + a(x - l)^4 + b(x - l)^3 + c(x - l)^2 + d
Final Prediction: Puzzle #115: Solve this question with explanation of the intermediate steps. Your response m ust 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 = E Correct Answer = D