Question

the polynomial p(x)=x⁴-2x³+3x²-mn+n when divided by (x-1)and (x+1)leaves the remainder 5 aand 19 . respectively find the value of a and b.hence find the remainder when p(x)is divided by (x-4)​

Answer 1

Answer:

47

Step-by-step explanation:

When f(x) is divided by (x+1) and (x-1) , the remainders are 19 and 5 respectively .

∴ f(-1) = 19 and f(1) = 5

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

⇒ 1 +2 + 3 + a + b = 19

∴ a + b = 13 ------- (i)

Again , f(1) = 5

⇒ 14 - 2 × 13 + 3 × 12 - a × 1 b = 5

⇒ 1 - 2 + 3 - a + b = 5

∴ b - a = 3 ------ (ii)

solving eqn (i) and (ii) , we get

a = 5 and b = 8

Now substituting the values of a and b in f(x) , we get

∴ f(x) = x4 - 2x3 + 3x2 - 5x + 8

Now f(x) is divided by (x-3) so remainder will be f(3)

∴ f(x) = ∴ f(x) = x4 - 2x3 + 3x2 - 5x + 8

⇒ f(3) = 34 - 2 × 33 + 3 × 32 - 5 × 3 + 8

= 81 - 54 + 27 - 15 + 8 = 47