On dividing 2x³ + 2ax² - 5x + a by x + a find the remainder.

Let p(x) = 2x³ + 2ax² - 5x + a

We will find remainder using remainder theorem which states that

If a polynomial p(x) is divided by a polynomial x – c then the remainder is p(c)

x + a can be written as x – (-a)

compare x – (-a) with x – c we get c = -a

⇒ p(c) = p(-a)

Substitute -a in 2x³ + 2ax² - 5x + a

⇒ p(-a) = 2(-a)³ + 2a(-a)² – 5(-a) + a

⇒ p(-a) = -2a³ + 2a^{3 +} 5a + a

⇒ p(-a) = 6a

Hence remainder is 6a when 2x³ + 2ax² - 5x + a is divided by x + a