Question

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If f (x) = ${x}^4 -{2x}^3 + {3x}^2 - {ax} + {b}$ is a polynomial such that when it is divided by ( x - 1) and ( x + 1 ), the remainders are 5 and 19. Determine the remainder f (x) is divided by ( x - 2 )

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Full explanation please ♥ ​

Answer 1

f(x) = x⁴ - 2x³ + 3x² - ax + b

_________ [GIVEN]

When the above polynomial is divided by (x - 1) and (x + 1) the remainders are 5 and 19.

• x - 1 = 0

=> x = + 1

When x = 1. It's remainder is 5.

» Similarly

• x + 1 = 0

=> x = - 1

When x = - 1. It's remainder is 19.

• A.T.Q.

f(x) = x⁴ - 2x³ + 3x² - ax + b

=> f(1) = (1)⁴ - 2(1)³ + 3(1)² - a(1) + b = 5

=> 1 - 2 + 3 - a + b = 5

=> 2 - a + b = 5

=> - a + b = 5 - 2

=> - a + b = 3

=> b = 3 + a _________(eq 1)

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» Similarly

=> f(-1) = (-1)⁴ - 2(-1)³ + 3(-1)² - a (-1) + b = 19

=> 1 + 2 + 3 + a + b = 19

=> 6 + a + b = 19

=> a + b = 19 - 6

=> a + b = 13

=> a + (3 + a) = 13 [From (eq 1)]

=> 2a + 3 = 13

=> 2a = 10

=> a = 5

• Put value of a in (eq 1)

=> b = 3 + 5

=> b = 8

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☞ Also it is said that if the polynomial is divided by (x - 2). Then what is it's remainder.

x - 2 = 0

=> x = + 2

=> f(2) = (2)⁴ - 2(2)³ + 3(2)² - a(2) + b

=> 16 - 16 + 12 - 2a + b

=> 12 - 2a + b

• Put value of a and b in above equation.

=> 12 - 2(5) + 8

=> 12 - 10 + 8

=> 10

_________________ [ANSWER]