Question

when f(y)=y⁴-4y³+8y²-my+n divided by y+1 and y-1 , we get the remainder of 10 and 16 respectively. find the remainder if f(y) is divided by y-3

Answer 1

Answer:

Step-by-step explanation:

Answer 2

The answer is 70

Given: f(y) =y⁴-4y³+8y²-my+n

When f(y) is divided by (y+1) reminder is 10

When f(y) is divided by (y-1) reminder is 16  

To find: The remainder when f(y) is divided by (y-3)

Solution:

When f(y) is divided by (y+1) reminder is 10

If  y+1 = 0 ⇒ y = -1 then  f(-1) will be equal to 10  

⇒ f(-1) = (-1)⁴-4(-1)³+8(-1)²-m(-1)+n = 10  

⇒ 1 - 4(-1) + 8 (1) + m + n = 10

⇒ 13 + m + n =10

⇒ m+n = - 3 _ (1)

When f(y) divided by (y+2) reminder is 16

If y - 1 = 0 ⇒ y = 1 then f(1) = 16

⇒ f(1) = (1)⁴-4(1)³+8(1)²-m(1)+n = 16

⇒ 1 - 4 + 8 - m + n = 16

⇒ 5 - m + n = 16

⇒ - m + n = 11 _(2)

Add (1) and (2) ⇒  m + n - m + n = - 3 + 11

⇒  2n = 8 ⇒ n = 4

Substitute n = 4 in (1) ⇒ m + 4 = - 3  ⇒ m = -7    

f(y) = y⁴-4y³+8y²-(-7)y+4

⇒ f(y) =  y⁴- 4y³+8y²+7y+4  

When f(y) is divided by (y -3)

⇒ y-3 = 0 ⇒ y = 3 then remainder is f(3)

f(3) =  (3)⁴- 4(3)³+8(3)²+7(3)+4  = 81 - 4(27)+(8(9)+21+ 4

     = 81 - 108 + 72 +21 +4 = 70

Therefore, the reminder is 70

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