Question

the polynomial 2x^2+3x+1 devided by x+2 leave remainder
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Answer 1

Given polynomial:

p(x) = 2x³ + 3x² + ax + b

when p(x) is divided by x - 2  leaves remainder = 2

when p(x) is divided by x + 2  leaves remainder = -2

To find: Value of a and b

We use remainder theorem.

Remainder Theorem states that if a polynomial p(x) is divided by a polynomial of form x - a then the remainder = p(a).

So,

when p(x) is divided by x - 2 ,

Remainder = p(2)

2(2)³ + 3(2)² + a(2) + b = 2

16 + 12 + 2a + b = 2

2a + b = 2 - 28

2a + b = -26 ...............................(1)

when p(x) is divided by x + 2 ,

Remainder = p(2)

2(-2)³ + 3(-2)² + a(-2) + b = -2

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

-2a + b = -2 + 4

-2a + b = 2 ...............................(2)

Add (1) and (2)

b + b = -26 + 2

2b = -24

b = -12

Put value of b in (1)

2a + (-12) = -26

2a = -26 + 12

2a = -14

a = -7

Therefore, Value of a is -7 and Value of b is -12

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