Question

An unknown polynomial yields a reminder 2 upon division by x-1, and remainder 1 upon division by x-2. What remainder is obtained if this polynomial is divided by {x-1}{x-2}.

Answer 1

Dear Student,

Please find below the solution to the asked query:
P ÷ (x – 1) remainder 2
P ÷ (x – 2) remainder 1
the difference in divisisor and remainder is
first case (x – 1) – 2 = (x – 3)
second case (x – 2) – 1 = (x – 3)
is same
so least polynomial is
(LCM of divisors) – difference
= (x – 1)(x – 2) – (x – 3)
= x^2 – 3x + 2 – x + 3
= (x^2 – 4x + 5)

so remainder when
(x^2 – 4x + 5) ÷ (x – 1)(x – 2)
= (x^2 – 4x + 5) ÷ (x^2 – 3x + 2)
quotient 1 and remainder
= (x^2 – 4x + 5) – (x^2 – 3x + 2)
= – x + 3
= 3 – x
Answer

Hope this information will clear your doubts about the topic.