Factored form to roots

A question is this sub-type if and only if the polynomial is already given in factored form (product of quadratics) and asks to find all roots by solving each quadratic factor.

7 questions · Moderate -0.7

4.02k Argand diagrams: geometric interpretation
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Edexcel F1 2018 June Q5
8 marks Moderate -0.3
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.
Edexcel FP1 2013 January Q5
7 marks Moderate -0.8
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.
Edexcel FP1 2009 June Q3
7 marks Moderate -0.8
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.
OCR FP1 2007 January Q5
7 marks Moderate -0.8
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 \(z ^ { 3 } - 8 = 0\).
Edexcel F1 2021 June Q5
7 marks Moderate -0.3
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}
Pre-U Pre-U 9794/2 2014 June Q7
2 marks Moderate -0.8
7
  1. Express \(z ^ { 4 } + 3 z ^ { 2 } - 4\) in the form \(\left( z ^ { 2 } + a \right) \left( z ^ { 2 } + b \right)\) where \(a\) and \(b\) are real constants to be found.
  2. Hence draw an Argand diagram showing the points that represent the roots of the equation \(z ^ { 4 } + 3 z ^ { 2 } - 4 = 0\).
Edexcel FP1 2013 June Q4
6 marks Moderate -0.8
$$f(x) = (4x^2 + 9)(x^2 - 2x + 5)$$
  1. Find the four roots of \(f(x) = 0\) [4]
  2. Show the four roots of \(f(x) = 0\) on a single Argand diagram. [2]