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Let P(x) be a polynomial with positive integer coefficients and P(0) > 1 The product of all the coefficients is 47 and the remainder when P(x) is divided by (x - 1) is 9887. What is the degree of P(x) ? (1) 9887 (2) 9840 (3) 9934 (4) 4920?
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Let P(x) be a polynomial with positive integer coefficients and P(0) >...
**Problem Analysis**
We are given that the polynomial P(x) has positive integer coefficients and P(0) > 1. We are also given that the product of all the coefficients is 47 and the remainder when P(x) is divided by (x - 1) is 9887. We need to determine the degree of P(x).
**Solution**
Since the product of all the coefficients is 47, and the coefficients are positive integers, we can deduce that P(x) has only two coefficients: 1 and 47. This is because the product of two positive integers can only be 47 if one of them is 1 and the other is 47.
Let's consider the polynomial P(x) = 47x^n + 1, where n is the degree of P(x).
**Remainder when P(x) is divided by (x - 1)**
To find the remainder when P(x) is divided by (x - 1), we can use the Remainder Theorem. According to the Remainder Theorem, the remainder when P(x) is divided by (x - 1) is equal to P(1).
Substituting x = 1 in P(x) = 47x^n + 1, we get P(1) = 47(1)^n + 1 = 48.
Therefore, the remainder when P(x) is divided by (x - 1) is 48.
But we are given that the remainder is 9887. This means that n cannot be a positive integer because 48 is not equal to 9887.
**Conclusion**
Since n cannot be a positive integer, P(x) must be a constant polynomial. The degree of a constant polynomial is always 0.
Therefore, the degree of P(x) is 0.
Answer: (4) 4920
We are given that the polynomial P(x) has positive integer coefficients and P(0) > 1. We are also given that the product of all the coefficients is 47 and the remainder when P(x) is divided by (x - 1) is 9887. We need to determine the degree of P(x).
**Solution**
Since the product of all the coefficients is 47, and the coefficients are positive integers, we can deduce that P(x) has only two coefficients: 1 and 47. This is because the product of two positive integers can only be 47 if one of them is 1 and the other is 47.
Let's consider the polynomial P(x) = 47x^n + 1, where n is the degree of P(x).
**Remainder when P(x) is divided by (x - 1)**
To find the remainder when P(x) is divided by (x - 1), we can use the Remainder Theorem. According to the Remainder Theorem, the remainder when P(x) is divided by (x - 1) is equal to P(1).
Substituting x = 1 in P(x) = 47x^n + 1, we get P(1) = 47(1)^n + 1 = 48.
Therefore, the remainder when P(x) is divided by (x - 1) is 48.
But we are given that the remainder is 9887. This means that n cannot be a positive integer because 48 is not equal to 9887.
**Conclusion**
Since n cannot be a positive integer, P(x) must be a constant polynomial. The degree of a constant polynomial is always 0.
Therefore, the degree of P(x) is 0.
Answer: (4) 4920
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Let P(x) be a polynomial with positive integer coefficients and P(0) > 1 The product of all the coefficients is 47 and the remainder when P(x) is divided by (x - 1) is 9887. What is the degree of P(x) ? (1) 9887 (2) 9840 (3) 9934 (4) 4920?
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Let P(x) be a polynomial with positive integer coefficients and P(0) > 1 The product of all the coefficients is 47 and the remainder when P(x) is divided by (x - 1) is 9887. What is the degree of P(x) ? (1) 9887 (2) 9840 (3) 9934 (4) 4920? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about Let P(x) be a polynomial with positive integer coefficients and P(0) > 1 The product of all the coefficients is 47 and the remainder when P(x) is divided by (x - 1) is 9887. What is the degree of P(x) ? (1) 9887 (2) 9840 (3) 9934 (4) 4920? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let P(x) be a polynomial with positive integer coefficients and P(0) > 1 The product of all the coefficients is 47 and the remainder when P(x) is divided by (x - 1) is 9887. What is the degree of P(x) ? (1) 9887 (2) 9840 (3) 9934 (4) 4920?.
Let P(x) be a polynomial with positive integer coefficients and P(0) > 1 The product of all the coefficients is 47 and the remainder when P(x) is divided by (x - 1) is 9887. What is the degree of P(x) ? (1) 9887 (2) 9840 (3) 9934 (4) 4920? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about Let P(x) be a polynomial with positive integer coefficients and P(0) > 1 The product of all the coefficients is 47 and the remainder when P(x) is divided by (x - 1) is 9887. What is the degree of P(x) ? (1) 9887 (2) 9840 (3) 9934 (4) 4920? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let P(x) be a polynomial with positive integer coefficients and P(0) > 1 The product of all the coefficients is 47 and the remainder when P(x) is divided by (x - 1) is 9887. What is the degree of P(x) ? (1) 9887 (2) 9840 (3) 9934 (4) 4920?.
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Let P(x) be a polynomial with positive integer coefficients and P(0) > 1 The product of all the coefficients is 47 and the remainder when P(x) is divided by (x - 1) is 9887. What is the degree of P(x) ? (1) 9887 (2) 9840 (3) 9934 (4) 4920?, a detailed solution for Let P(x) be a polynomial with positive integer coefficients and P(0) > 1 The product of all the coefficients is 47 and the remainder when P(x) is divided by (x - 1) is 9887. What is the degree of P(x) ? (1) 9887 (2) 9840 (3) 9934 (4) 4920? has been provided alongside types of Let P(x) be a polynomial with positive integer coefficients and P(0) > 1 The product of all the coefficients is 47 and the remainder when P(x) is divided by (x - 1) is 9887. What is the degree of P(x) ? (1) 9887 (2) 9840 (3) 9934 (4) 4920? theory, EduRev gives you an
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