If \(x^2 - 4x + 3\) is a factor of \(x^4 + px^2 + q\), then what is \((p - q)\) equal to?
\(-19\)
\(x^2-4x+3=(x-1)(x-3)\). Dividing \(x^4+px^2+q\) by \(x^2-4x+3\) gives quotient \(x^2+4x+(p+13)\) with remainder \([4(p+13)-12]x+[q-3(p+13)]\). For exact division both must vanish: \(4(p+13)-12=0 \Rightarrow p=-10\), and \(q=3(p+13)=9\). So \((x^2-4x+3)(x^2+4x+3)=x^4-10x^2+9\), confirming \(p=-10, q=9\). Hence \(p-q=-10-9=-19\).
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