If dividing P(x) = 4x³ - 7x² + bx - 5 by (x - 3) results the remainder 10, then find the value of b.

Question: If dividing P(x) = 4x³- 7x² + bx - 5 by (x - 3) results the remainder 10, then find the value of b.

Solution:
According to the Remainder Theorem, if a polynomial P(x) is divided by (x - c), then the remainder = P(c).
Here divisor is (x - 3).
So remainder = P(3).

Now,
P(3) = 4(3)³ - 7(3)² + b(3) - 5

= 4 × 27 - 7 × 9 + 3b - 5

= 108 - 63 + 3b - 5

= 40 + 3b

According to the question, the remainder is 10.
So, 40 + 3b = 10

⇒ 3b = 10 - 40
⇒ 3b = - 30
⇒ b = - 10