Find the values of a and b, if $x^{2} - 4$ is a factor of $a x^{4} + 2 x^{3} - 3 x^{2} + b x - 4$

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Solution

$\begin{matrix} i f x^{2} - 4 i s a f a c t o r o f a x^{4} - + 2 x^{3} 3 x^{2} + b x - 4 x^{2} - 4 = 0 x = \pm 2 a r e z e r o e s o f t h e p o l y n o m i a l p (x) = a x^{4} + 2 x^{3} - 3 x^{2} + b x - 4 ∴ p (2) = 0 a n d p (- 2) = 0 p (2) = 0 t h e n 16 a + 16 - 12 + 2 b - 4 = 0 16 a + 2 b = 0....... (1) p (- 2) = 0 t h e n 16 a - 16 - 12 - 2 b - 4 = 0 16 a - 2 b = 32...... (2) s u b s t r a c t (2) f r o m (1) w e g e t 4 b = - 32 b = - 8 s u b s t i t u d e b i s e q u a t i o n s (1) w e g e t 16 a - 16 = 0 ∴ a = 1 a n d b = - 8 \end{matrix}$

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Respected Sir,

Please help me in solving my below mentioned doubt:

Find the values of a and b, if x² - 4 is a factor of ax⁴ + 2x³ - 3x² + bx - 4.

{(a x)}^{4} + 2 x - {3 x}^{2} + b x - 4

x^{2} - 4

2 x^{3} + a x^{2} + b x + 12

Find the value of a for which (x - 4) is a factor of $(2 x^{3} - 3 x^{2} - 18 x + a)$ .

2 x^{3} + a x^{2} + 2 b x + 1

2 a - 3 b = 4

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