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\includegraphics[alt={},max width=\textwidth]{6be6c0b0-76b7-49c0-bf1b-dc6f8f79981b-2_836_906_361_675}
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\caption{Fig. 12}
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Fig. 12 shows the graph of \(y = \frac { 4 } { x ^ { 2 } }\).
- On the copy of Fig. 12, draw accurately the line \(y = 2 x + 5\) and hence find graphically the three roots of the equation \(\frac { 4 } { x ^ { 2 } } = 2 x + 5\).
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[3] - Show that the equation you have solved in part (i) may be written as \(2 x ^ { 3 } + 5 x ^ { 2 } - 4 = 0\). Verify that \(x = - 2\) is a root of this equation and hence find, in exact form, the other two roots.
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[6] - By drawing a suitable line on the copy of Fig. 12, find the number of real roots of the equation \(x ^ { 3 } + 2 x ^ { 2 } - 4 = 0\).
- You are given that \(\mathrm { f } ( x ) = ( 2 x - 5 ) ( x - 1 ) ( x - 4 )\).
(A) Sketch the graph of \(y = \mathrm { f } ( x )\).
(B) Show that \(\mathrm { f } ( x ) = 2 x ^ { 3 } - 15 x ^ { 2 } + 33 x - 20\). - You are given that \(\mathrm { g } ( x ) = 2 x ^ { 3 } - 15 x ^ { 2 } + 33 x - 40\).
(A) Show that \(\mathrm { g } ( 5 ) = 0\).
(B) Express \(\mathrm { g } ( x )\) as the product of a linear and quadratic factor.
(C) Hence show that the equation \(\mathrm { g } ( x ) = 0\) has only one real root. - Describe fully the transformation that maps \(y = \mathrm { f } ( x )\) onto \(y = \mathrm { g } ( x )\).