Question

Find the remainder when f(x) = 12 x3 - 13x2 - 5x + 7 divided by (3x + 2)(2)
The polynomial f (x) = x4 - 2x3 + 3x? - ax + b when divided by (x-1) and (x + 1) leaves the
remainder 5 and 19 respectively. Find the value of a and b.
(3)
For what value of k is the polynomial (2x4 + 3x + 2kx2 + 3x + 6) is exactly divisible by (x + 2)?
Without actual division, prove that (2x4 - 6x2 + 3x2 + 3x - 2) is exactly divisible by (x2-3x + 2)
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Answer 1

Answer:

(1)since f(x) is divided by linier expression so remainder will be constant

Hence,

f(x)=(6x+4)h(x)+k

Put x=-2/3

f(-2/3)=k

k=73/9

(2) f(1) =5

b-a=3 - - - - - (1)

f(-1)=19

b+a=19 - - - - (2)

Solve( 1) and (2)

a=8,b=11.

(3)

For exactly divisibility by (x+2),

f(-2)=0

k=-1

(4)

x^2-3x+2=(x-2)(x-1)

Make it zero, get

x=1,2

Now check, f(1)

f(1)=0,also f(2)=0

Hence divisible by (x-1)(x-2).