IB AA HL May 2026 P1 TZA Q8

Consider the polynomial \(p(z)=3z^5-7z^4+cz^3+dz^2+ez-26\), where \(z\in\mathbb{C}\) and \(c,d,e\in\mathbb{R}\).

Three of the roots of \(p(z)\) are \(\frac13\), \(a-i\), \(2+bi\), where \(a,b\in\mathbb{R}\), \(b>1\).

1. By considering the sum and the product of the roots of \(p(z)\),
   1. show that \(a=-1\);
   2. find the value of \(b\).[5]
2. Show that \(p(z)=(Az+B)(z^2+Cz+D)(z^2+Ez+F)\), where \(A,B,C,D,E,F\) are integers to be determined.[3]