The function f is defined by
$$f ( z ) = z ^ { 4 } + 3 z ^ { 2 } - 6 z + 10 \quad z \in \mathbb { C }$$
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Show that (1+i) is a root of \(\mathrm { f } ( \mathrm { z } ) = 0\)
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(ii) Hence write down another root of \(\mathrm { f } ( \mathrm { z } ) = 0\)
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(iii) One of the linear factors of \(\mathrm { f } ( \mathrm { z } )\) is
$$( z - ( 1 + i ) )$$
Write down another linear factor and hence, or otherwise, find a quadratic factor of \(\mathrm { f } ( \mathrm { z } )\) with real coefficients.
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(iv) Find another quadratic factor of \(\mathrm { f } ( \mathrm { z } )\) with real coefficients.
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(v) Hence explain why the graph of \(y = \mathrm { f } ( x )\) does not intersect the \(x\)-axis.