If the polynomials x³ + 3x² – m and 2x³ – mx + 9 leaves the same remainder when they are divided by (x – 2), find the value of m. Also find the remainder

x³ + 3x² – m and 2x³ – mx + 9 is divided by (x – 2) and the remainder is same.

Now let p(x) = x³ + 3x² – m is divided by x – 2 then, p(2) is the remainder

p(2) = (2)³ + 3(2)² – m

= 8 + 12 – m

= 20 – m

Now let q(x) = 2x³ – mx + 9 is divided by x – 2 then, q(2) is the remainder

q(2) = 2(2)³ – m(2) + 9

= 16 – 2m + 9

= 25 – 2m

Now, as the question says that the remainder for p(x) and q(x) is same

Therefore, p(2) = q(2)

20 – m = 25 – 2m

2m – m = 25 – 20

m = 5

Remainder = p(2) = 20 – m

= 15