Question

When the polynomial 2x^3+3x^2+ax+b is divided by (x-2) leaves remainder 2 and x+2 leaves remainder-2.Find a and b?

Answer 1

${\huge {\mathfrak {Answer :-}}}$

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p(x) = 2x^3 + 3x^2 + ax + b

Now,

p(2) = 2 * 8 + 3 * 4 + 2a + b = 2

Or, 16 + 12 + 2a + b = 2

Or, 2a + b = - 26

And,

p(-2) = 2 * (-8) + 3 * 4 + ( - 2)a + b = - 2

Or, - 16 + 12 - 2a + b = - 2

Or, - 2a + b = 2

From the equations,

2a + b - 2a + b = - 26 + 2

Or, 2b = - 24

Or, b = - 12

And,

2a + (-12) = - 26

Or, 2a = - 14

Or, a = - 7

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Thanks..

Answer 2

p(x) = 2x³ + 3x² + ax + b

When the above polynomial is divided by (x - 2) it leaves a remainder 2 and when it is divided by (x + 2) it leaves a remainder -2.

___________ [ GIVEN ]

We have to find a and b.

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p(x) = 2x³ + 3x² + ax + b

• If x = 2

=> p(2) = 2(2)³ + 3(2)² + a(2) + b

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

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

=> 2a + b = - 26 _______ (eq 1)

• If x = - 2

=> p(-2) = 2(-2)³ + 3(-2)² + (-2)a + b

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

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

=> - 2a + b = 2 ________ (eq 2)

Add (eq 1) and (eq 2)

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

=> 2b = - 24

=> b = - 12

Put value of of b in (eq 1)

=> 2a + (-12) = - 26

=> 2a - 12 = -26

=> 2a = - 26 + 12

=> 2a = - 14

=> a = - 7

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a = - 7 and b = - 12

_________ [ ANSWER ]

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✡ Verification :

We have equation :

- 2a + b = 2

From above calculations we have a = -7 and b = -12

Put values of a and b in above equation

→ - 2(-7) -12 = 2

→ 14 - 12 = 2

→ 2 = 2

We get the same result when we use the equation :

2a + b = -26

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