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The polynomial f(x)= x -2x 3x² - ax b when divided by (x − 1) and (x 1) leaves the remainders 5 and 19 respectively. Find the values of a and b. Hence, find the remainder when f(x) is divided by (x-3).?
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The polynomial f(x)= x -2x 3x² - ax b when divided by (x − 1) and (x...
Solution:
Given,
f(x) = x - 2x + 3x² - ax + b
When f(x) is divided by (x - 1), the remainder is 5.
f(1) = 5
f(x) = (x - 1)q(x) + 5
When f(x) is divided by (x + 1), the remainder is 19.
f(-1) =19
f(x) = (x + 1)p(x) + 19
Finding the values of a and b:
To find the values of a and b, we need to solve the two equations obtained from the given conditions.
f(x) = (x - 1)q(x) + 5
f(x) = (x + 1)p(x) + 19
Substituting f(x), we get:
(x - 1)q(x) + 5 = (x + 1)p(x) + 19
Expanding the terms, we get:
xq(x) - q(x) + 5 = xp(x) + p(x) + 19
Equating the coefficients of x, we get:
q(x) = p(x)
Equating the constant terms, we get:
5 = 19
This is not possible.
Therefore, there are no values of a and b that satisfy the given conditions.
Finding the remainder when f(x) is divided by (x - 3):
To find the remainder when f(x) is divided by (x - 3), we can use the remainder theorem.
f(3) = 3 - 2(3) + 3(3)² - a(3) + b
f(3) = -9a + b + 24
Therefore, the remainder when f(x) is divided by (x - 3) is -9a + b + 24.
Given,
f(x) = x - 2x + 3x² - ax + b
When f(x) is divided by (x - 1), the remainder is 5.
f(1) = 5
f(x) = (x - 1)q(x) + 5
When f(x) is divided by (x + 1), the remainder is 19.
f(-1) =19
f(x) = (x + 1)p(x) + 19
Finding the values of a and b:
To find the values of a and b, we need to solve the two equations obtained from the given conditions.
f(x) = (x - 1)q(x) + 5
f(x) = (x + 1)p(x) + 19
Substituting f(x), we get:
(x - 1)q(x) + 5 = (x + 1)p(x) + 19
Expanding the terms, we get:
xq(x) - q(x) + 5 = xp(x) + p(x) + 19
Equating the coefficients of x, we get:
q(x) = p(x)
Equating the constant terms, we get:
5 = 19
This is not possible.
Therefore, there are no values of a and b that satisfy the given conditions.
Finding the remainder when f(x) is divided by (x - 3):
To find the remainder when f(x) is divided by (x - 3), we can use the remainder theorem.
f(3) = 3 - 2(3) + 3(3)² - a(3) + b
f(3) = -9a + b + 24
Therefore, the remainder when f(x) is divided by (x - 3) is -9a + b + 24.
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The polynomial f(x)= x -2x 3x² - ax b when divided by (x − 1) and (x 1) leaves the remainders 5 and 19 respectively. Find the values of a and b. Hence, find the remainder when f(x) is divided by (x-3).?
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The polynomial f(x)= x -2x 3x² - ax b when divided by (x − 1) and (x 1) leaves the remainders 5 and 19 respectively. Find the values of a and b. Hence, find the remainder when f(x) is divided by (x-3).? for Class 9 2024 is part of Class 9 preparation. The Question and answers have been prepared according to the Class 9 exam syllabus. Information about The polynomial f(x)= x -2x 3x² - ax b when divided by (x − 1) and (x 1) leaves the remainders 5 and 19 respectively. Find the values of a and b. Hence, find the remainder when f(x) is divided by (x-3).? covers all topics & solutions for Class 9 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The polynomial f(x)= x -2x 3x² - ax b when divided by (x − 1) and (x 1) leaves the remainders 5 and 19 respectively. Find the values of a and b. Hence, find the remainder when f(x) is divided by (x-3).?.
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