Factored form to roots

5. Given that $$z ^ { 4 } - 6 z ^ { 3 } + 34 z ^ { 2 } - 54 z + 225 \equiv \left( z ^ { 2 } + 9 \right) \left( z ^ { 2 } + a z + b \right)$$ where \(a\) and \(b\) are real numbers,

1. find the value of \(a\) and the value of \(b\).
2. Hence find the exact roots of the equation $$z ^ { 4 } - 6 z ^ { 3 } + 34 z ^ { 2 } - 54 z + 225 = 0$$
3. Show your roots on a single Argand diagram.

5. $$f ( x ) = \left( 4 x ^ { 2 } + 9 \right) \left( x ^ { 2 } - 6 x + 34 \right)$$

1. Find the four roots of \(\mathrm { f } ( x ) = 0\) Give your answers in the form \(x = p + \mathrm { i } q\), where \(p\) and \(q\) are real.
2. Show these four roots on a single Argand diagram.

3. $$f ( x ) = \left( x ^ { 2 } + 4 \right) \left( x ^ { 2 } + 8 x + 25 \right)$$

1. Find the four roots of \(\mathrm { f } ( x ) = 0\).
2. Find the sum of these four roots.

5

1. Verify that \(z ^ { 3 } - 8 = ( z - 2 ) \left( z ^ { 2 } + 2 z + 4 \right)\).
2. Solve the quadratic equation \(z ^ { 2 } + 2 z + 4 = 0\), giving your answers exactly in the form \(x + \mathrm { i } y\). Show clearly how you obtain your answers.
3. Show on an Argand diagram the roots of the cubic equation \(z ^ { 3 } - 8 = 0\).

5. $$f ( x ) = \left( 9 x ^ { 2 } + d \right) \left( x ^ { 2 } - 8 x + ( 10 d + 1 ) \right)$$ where \(d\) is a positive constant.

1. Find the four roots of \(\mathrm { f } ( x )\) giving your answers in terms of \(d\). Given \(d = 4\)
2. Express these four roots in the form \(a + \mathrm { i } b\), where \(a , b \in \mathbb { R }\).
3. Show these four roots on a single Argand diagram. \includegraphics[max width=\textwidth, alt={}, center]{d7689f4a-a41e-45be-911b-4a74e81997eb-21_2647_1840_118_111}

11. (a) Using that 3 is the real root of the cubic equation \(x ^ { 3 } - 27 = 0\), show that the complex roots of the cubic satisfy the quadratic equation \(x ^ { 2 } + 3 x + 9 = 0\).
(b) Hence, or otherwise, find the three cube roots of 27, giving your answers in the form \(a + \mathrm { i } b\), where \(a , b \in \mathbb { R }\).
(c) Show these roots on an Argand diagram.