The number of values $a$ can take such that $x^4 + ax^3 + (3a - 4)x^2 + 2(a - 1)x - 4$ can be expressed as a product of two quadratic polynomials, $x^2 + px + 2$ and $x^2 + qx - 2$, where…

Question

The number of values $a$ can take such that $x^4 + ax^3 + (3a - 4)x^2 + 2(a - 1)x - 4$ can be expressed as a product of two quadratic polynomials, $x^2 + px + 2$ and $x^2 + qx - 2$, where $p$ and $q$ are real, is

Accepted Answer

B - 1. (x^2+px+2)(x^2+qx-2) = x^4 + (p+q)x^3 + pq x^2 + 2(q-p)x - 4. So p+q = a, q-p = a-1, giving p = 1/2, q = a - 1/2; then pq = 3a-4 gives (2a-1)/4 = 3a-4, a = 3/2. Only one value.

Answer

A - 2

Answer

C - 0

Answer

D - 3

The number of values \(a\) can take such that \(x^4 + ax^3 + (3a - 4)x^2 + 2(a - 1)x - 4\) can be expressed as a product of two quadratic polynomials, \(x^2 + px + 2\) and \(x^2 + qx - 2\), where…

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