If dividing P(x) = 3x³ - 5x² + cx - 7 by (x - 2) results in the remainder 5, then find the value of c.

Question: If dividing P(x) = 3x³ - 5x² + cx - 7 by (x - 2) results in the remainder 5, then find the value of c.

Solution:
According to the Remainder Theorem, if a polynomial P(x) is divided by (x - k), then the remainder = P(k).
Here, the divisor is (x - 2), so remainder = P(2).

Now,
P(2) = 3(2)³ - 5(2)² + c(2) - 7
= 3 × 8 - 5 × 4 + 2c - 7
= 24 - 20 + 2c - 7
= (24 − 20 - 7) + 2c
= - 3 + 2c

According to the question, the remainder is 5.
So,
- 3 + 2c = 5
⇒ 2c = 5 + 3
⇒ 2c = 8
∴ c = 4