If (x³ + mx² – x + 6) has (x - 2) as a factor and leaves a remainder r, when divided by (x - 3), find the values of m and r.

If (x - 2) is a factor of the polynomial (x³ + mx² – x + 6) then it must satisfy it.

So, putting x = 2 the polynomial must be zero.

Putting x = 2 and equating to zero.

= (23 + m2² –2 + 6)

= 4m + 12 = 0

= m = -3

If we divide f(x) = (x³ + mx² –x + 6) by (x - 3) remainder can be find at value of –

(x - 3) = 0

Or x = 3

So we will put x = 3 in f(x) = (x³ + mx² – x + 6)

f(3) = (3³ + m3² – 3 + 6)

= 30 + 9m

So remainder = 30 + 9m

= 30 + 9(-3) = 30 - 27 = 3

So, r = 3.