Dividing a polynomial

Algebra Level 3

When a polynomial P ( x ) P(x) of degree n > 2 n > 2 is divided by x − 3 x-3 , the remainder is 5. When P ( x ) P(x) is divided by x − 1 x-1 , the remainder is 1. Find the remainder when P ( x ) P(x) is divided by x 2 − 4 x + 3 x^2-4x+3 .

1 1 2 x + 1 2x+1 2 x − 1 2x-1 3 3

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Teja Gupta by Teja Gupta , 22, USA

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1 solution

P ( x ) = ( x − 1 ) h ( x ) + 1 P(x) = (x-1)h(x) + 1
Put x = 1 , → P ( 1 ) = 1 x = 1, \to P(1) = 1
P ( x ) = ( x − 3 ) l ( x ) + 5 P(x) = (x-3)l(x) + 5
Put x = 3 , → P ( 3 ) = 5 x = 3, \to P(3) = 5

Since the divisor is a quadratic, the maximum degree of remainder is 1.
∴ P ( x ) = ( x 2 − 4 x + 3 ) g ( x ) + A x + B \therefore P(x) = (x^{2}-4x+3)g(x) + Ax+ B .
P ( 1 ) = 1 = A + B P(1) = 1 = A+B
P ( 3 ) = 5 = 3 A + B P(3) = 5 = 3A+B
Solving this system of linear equations we get, A = 2 , B = − 1 A = 2, B = -1
Thus the remainder is A x + B = 2 x − 1 Ax+B = 2x-1

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