Question

When a 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

$\huge\underline\green {\tt Answer:}$ a = - 23, b = - 44

$\textbf {\underline{\underline{Step\: - by\:- step\: Explanation\:}}}$

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i) g(x) = x - 2

=> x - 2 = 0 ( to find zero of the polynomial )

=> x = 2

P ( x ) = ${2x}^3 + {3x}^2 + {ax} + {b} = 2$

P ( 2 ) = $2 {(2)}^3 + 3 {(2)}^2 + a (2) + b = 2$

==> 2(8) + 3 (4) + 18 + b = 2

==> 16 + 12 + 18 + b = 2

==> 46 + b = 2

==> 2 = 46 + b

==> 2 - 46 = b

==> b = - 44

ii) g(x) = ( x + 2 )

==> x + 2 = 0

==> x = - 2

Put the value :

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

==> - 4 - 2a - 44 = - 2

==> - 48 - 2a = -2

==> - 2a = -2 + 48

==> - 2a = 46

==> a = $\dfrac {46}{-2}$

==> a = - 23

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BE BRAINLY!