Question

When a polynomial 2x³ + 3x² + ax + b is divided by ( x - 2 ) leaves reaminder 2 and ( x + 2 ) leave reaminder -2 . Find a and b .​

Answer 1

Step-by-step explanation:

Let

P(x) = 2x³ + 3x² + ax + b

If P(x) divided by (x-2) then the reaminder P(2) = 2

P(2) = 2(2)³ + 3(2)² + a(2) + b = 2

===> 2.8 + 3.4 + 2a + b = 2

===> 16 + 12 + 2a + b = 2

===> 28 + 2a + b = 2

===> 2a + b = 2 - 28

===> 2a + b = - 26 ===> eqn 1

If P(x) divided by (x+2) then the reaminder P(-2) = -2

P(-2) = 2(-2)³ + 3(-2)³ + a(-2) + b = -2

===> 2(-8) + 3(4) - 2a + b = -2

===> -16 + 12 - 2a + b = -2

===> - 4 - 2a + b = -2

===> -2a + b = -2 + 4

===> -2a + b = 2 ===> eqn 2

from eqn 1 and 2

2a + b = -26

-2a + b = +02

______________

00 + 2b = -24

====> b = -24 / 2

====> b = -12

Substitute "b" value in eqn1

2a + ( -12 ) = -26

2a = -26 + 12

2a = -14

a = 14 / 2

a = -7 and b = -12

Answer 2

Step-by-step explanation:

$\huge\bold\red{ Let }$

P(x) = 2x³ + 3x² + ax + b

If P(x) divided by (x - 2) then the remainder

P(2) = 2(2)³ + 3(2)² + a(2) + b = 2

• 2*16 + 3*4 + 2a + b = 2
• 16 + 12 + 2a + b = 2
• 28 + 2a + b = 2
• 2a + b = 2 - 28
• $\sf\fbox\pink{ 2a~ + ~b ~= ~-26 }$ ===> eqn ①

If P(x) divided by (x + 2) then the remainder P(-2) = 2(-2) ³ + 3(-2) ² + a(-2) + b = 2

• 2(-8) + 3(4) - 2a + b = -2
• - 16 + 12 - 2a + b = -2
• - 4 - 2a + b = -2
• - 2a + b = - 2 + 4
• $\sf\fbox\purple{2a ~+ ~b~ =~ 2}$ ===> eqn ②

　　　　From eqn ① and ②

• 2a + b = - 26
• - 2a + b = + 02
• ____________
• 00 + 2b = -24
• b = $\frac{ -24 }{ 2 }$
• b = - 12

Substitute "b" value in eqn ①

• 2a + (-12) = - 26
• 2a - 12 = - 26
• 2a = - 26 + 12
• 2a = - 14
• a = $\frac{ -14 }{ 2 }$
• a = -7

Hence,

• a = -7 and b = -12