Question

PLEASE ANSWER THIS QUESTION..​

Answer 1

Step-by-step explanation:

2).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

3)(x-1) and (x+1) are factors of the given polynomial.

x-1 = 0. ; x+1 = 0

x = 1. ; x = -1

Put x= -1,1 to find a and b

First,put x = 1

ax³+x²-2x+b = 0

a(1)³+(1)²-2(1)+b=0

a+1-2+b=0

a+b-1 = 0

a+b = 1 --------(1)

Put x = -1

a(-1)³+(-1)²-2(-1)+b = 0

a(-1)+1+2+b = 0

-a+b+3 = 0

-a+b = -3 -------(2)

(1)+(2)

a+b = 1

-a+b=-3

-----------

2b = -2

b = -2/2

b = -1

Put b= -1 in (1)

a+(-1) = 1

a-1 = 1

a=1+1

a=2

Therefore, a = 2 and b = -1