Question

find the values of a and b if
${x}^{2} - 4 \: is \: a \: factor \: of \: a {x}^{4} \: + 2 {x}^{3} \: - \: 3 {x}^{2} + \: bx \: - 4$
class 9

Answer 1

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Let p(x) = ax⁴ + 2x³ - 3x² + bx - 4

Factor => x² - 4

Thus x² - 4 = 0

=> x² = 4

=> x = +- √4

=> x = 2 or x = - 2

Thus p(2) = 0

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

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

=> 16a + 2b = 0

=> 8a + b = 0

=> b = - 8a

Also p(-2) = 0

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

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

=> 16a - 2b = 8

=> 8a - b = 4

=> 8a + 8a = 4 (b = - 8a)

=> 16a = 4

=> a = 1/4 <<<<<<< Answer

Thus b = -8/4

=> b = - 2 <<<<<<< Answer

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Hope this helps ✌️