A polynomial f(x) leaves remainder 15 when divided by (x3) and (2x+1)(x3) and (2x+1) when divided by (x1)2(x1)2when if is divided by (x3)(x1)2(x3)(x1)2 then remainder is
Answers
Answer:
Step-by-step explanation:
Let q(x) and r(x) be the quotient and remainder respectively got when the polynomial p(x) is divided by g(x) = (x-3)(x-1)^2.
Since the divisor g(x) is of third degree, the degree of the remainder r(x) is < or equal to 2. Hence, we can take r(x) = ax^2 + bx + c.
It is given p(3) = 15. So 9a+3b+c = 15…….(1)
When we divide the polynomial p(x) by (x-1)^2 , we will be continuing the division till we get a remainder whose degree is < or = 1. Hence the above remainder r(x) has to be further divided if we are performing division by (x-1)^2 only. The new remainder obtained when ax^2+bx+c is divided by (x-1)^2 = x^2 -2x +1 will be (2a+b)x + c-a. This is given to be 2x+1. Hence, 2a+b=2 or b= 2–2a, and c-a =1 gives c= a+1. Hence by substitution in (1) above, we have 9a + 2–2a + a+1 = 15 or a = 2 and so b= -2 and c= 3. Therefore, the remainder when the polynomial is divided by (x-3)(x-1)^2 will be 2x^2–2x+3.