Question

if f(y) = y⁴ -4y³ + 8y² -my +n is divisible by y+1 and y -1 leaves a remainder 10 and 16 respectively.. find the remainder when f(y) is divided by y-3​

Answer 1

Case-1:

f(y) is divided by y+1, and leaves remainder 10.

By Remainder Theorem:

f(-1)=10

(-1)⁴-4(-1)³+8(-1)²+m+n = 10

1+4+8+m+n = 10

13+m+n = 10

m+n = -3 ... eq. 1

Case-2:

f(y) is divided by y-1, and leaves remainder 16.

By Remainder Theorem:

f(1)=16

1⁴-4+8-m+n = 16

1-4+8-m+n = 16

5-m+n = 16

-m+n = 11 ...eq. 2

From eqs. 1 and 2:-

11+m = n

m+11+m = -3

11+2m = -3

-7 = m

11-7 = n

4 = n

So, f(y) = y⁴-4y³+8y²+7y+4

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

f(3) = 3⁴-4×3³+8×3²+7×3+4

= 81-108+72+21+4

= 70