How many roots survive the substitution?

8 written marks

Consider \(x^4-(p+1)x^2+p=0\), where \(p\) is real.

1. Using \(u=x^2\), factorise the equation in \(u\).

   [2]

2. Solve the original equation when \(p=16\).

   [2]

3. For every real \(p\), state the number of distinct real solutions of the original equation.

   [4]

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