Question

14. Find the values of a and b, if x^2 - 4 is a factor of ax^4 + 2x^3 - 3x^2 + bx -4​

Answer 1

Answer:

a = 1 & b = - 8

Step-by-step explanation:

x^2 - 4 = x^2 - 2^2 = (x + 2)(x - 2)

If x + 2 and x - 2 are factors of ax^4 + 2x^3 - 3x^2 + bx - 4, value of this polynomial must be 0, for x = - 2 or x = 2.

For x = - 2:

= > a(-2)⁴ + 2(-2)³ - 3(-2)² + b(-2) - 4 = 0

= > a(16) + 2(-8) - 3(4) + b(-2) - 4 = 0

= > 16a - 16 - 12 - 2b - 4 = 0

= > 16a - 2b - 32 = 0

= > 8a - b - 16 = 0

= > b = 8a - 16 ... (1)

For x = 2:

= > a(2)⁴ + 2(2)³ - 3(2)² + b(2) - 4 = 0

= > a(16) + 2(8) - 3(4) + b(2) - 4 = 0

= > 16a + 16 - 12 + 2b - 4 = 0

= > 16a + 2b = 0

= > 16a = - 2b

= > 8a = - b

= > 8a = - (8a - 16) { from (1) }

= > 8a = - 8a + 16

= > 8a + 8a = 16

= > a = 1

And then,

= > b = 8a - 16

= > b = 8(1) - 16 = - 8

Answer 2

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