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.

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Solution

if x²- 4 is a factor of ax⁴+ 2x³- 3x²+ bx - 4.

x² - 4 = 0 ⇒ x = ±2 are zeros of the polynomial p(x) = ax⁴+ 2x³- 3x²+ bx - 4.

∴ p(2) = 0 and p(-2) = 0.

p(2) = 0 then 16a + 16 -12 +2b -4 = 0

16a + 2b = 0 -------------(1)

p(-2) = 0 then 16a -16 -12 - 2b - 4 = 0

16a -2b = 32 -------------(2)

Substract (2) from (1) we get

4b = -32

b = -8

Substitute b is in equ(1) we get 16a - 16 = 0

∴ a = 1 and b = -8.

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