Question

If f(x) = x⁴ + px³ - 2x² + qx - 8 is divisible by x² - 4, find the values of p and q.

Answer 1

Answer:

x^2-4=0

x^2=4

therefore

X=+2 or -2(as -2*-2=4,2*2=4)

Answer 2

Answer:

Given f(x) = x⁴ + px³ - 2x² + qx - 8

x² - 4 = x² - 2² = (x+2)(x-2)

If f(x) is Divisible by (x+2) then the remainder is f(-2) = 0

=> (-2)⁴ + p(-2)³ - 2(-2²) + q × (-2) - 8 = 0

=> 16 - 8p - 8 - 2q - 8 = 0

=> -8p - 2q = 0

=> 4p + q = 0---(1)

If f(x) is Divisible by (x-2) then the remainder is f(2) = 0

=> 2⁴ + p(2³) - 2(2²) + q × 2 - 8 = 0

=> 16 + 8p - 8 + 2q - 8 = 0

=> 8p + 2q = 0

=> 4p + q = 0 ---(2)

For all real values of p and q which satisfies the eqution (1) and (2).

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