The polynomial f (x) leaves a remainder of - 3 and - 7 when divided by (3x - 1) and (x + 1) respectively.

Find the remainder when f (x) is divided by (3x^2 + 2x - 1).

Step-by-step explanation:

Here, f (x) is the given polynomial.

By remainder Theorem,

When divided by (3x-1),

f (1/3) = - 3 ... (1)

When divided by (x+1),

f (-1) = - 7 ... (2)

Another polynomial is 3x²+2x-1

Solving,

3x²+2x-1

= 3x²+3x-x-1

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

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

So

f (x) = (3x-1) (x+1) Qx + (ax+b)

For f (-1),

-7 = - a+b

b = a-7

For f (1/3),

-3 = a/3+b

or, - 3 = a/3+a-7

or, 4*3 = 4a

or a = 3

Also, b = 3-7 = -4

Hence, remainder is (3x-4)