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Question

If ax3+bx2+cx+d is divisible by x2+h2, prove that ad=bc.

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Solution

Let the cubic equation ax3+bx2+cx+d is divisible by x2+h2.

Then we can write ,

ax3+bx2+cx+d=(x2+h2)(px+q)

ax3+bx2+cx+d=px(x2+h2)+q(x2+h2)

ax3+bx2+cx+d=px3+qx2+ph2x+qh2

Therefore ,
upon comparing the both eqution we get ,

p=a
q=b
ph2=c
qh2=d

Thus ,
ad=pqh2
bc=pqh2

Thus ad=bc

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