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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.
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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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