Question

when a polynomial 2x³+3x²+ax+b is divided by (x-2) leaves remainder 2, and (x+2) leaves remainder -2.find a and b​

Answer 1

Answer:

P(x) = x⁴ - 2x³ + 3x² - ax + 3a - 7.

Divisor = x + 1

∴ x + 1 = 0

∴ x = -1

Thus,

P(-1) = (-1)⁴ - 2(-1)³ + 3(-1)² - a(-1) + 3a - 7.

19 = 1 + 2 + 3 + a + 3a - 7

19 = 6 - 7 + 4a

4a - 1 = 19

4a = 20

⇒ a = 4

∴ Value of a is 4.

Now, the Polynomial will be ---→

P(x) = x⁴ - 2x³ + 3x² - (4)x + 3(4) - 7

P(x) = x⁴ - 2x³ + 3x² - 4x + 12 - 7

P(x) = x⁴ - 2x³ + 3x² - 4x + 5

Now, When this polynomial is divided by (x + 2), then,

x + 2 = 0

x = - 2

∴ P(-2) = (-2)⁴ - 2(-2)³ + 3(-2)² - 4(-2) + 5

⇒ P(-2) = 16 + 16 + 12 + 8 + 5

⇒ P(-2) = 57