Sketch the curve with equation
$$y = x ^ { 2 } ( 2 x + a )$$
where \(a > 0\)
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6
The polynomial \(\mathrm { p } ( x )\) is given by
$$\mathrm { p } ( x ) = x ^ { 2 } ( 2 x + a ) + 36$$
6
It is given that \(x + 3\) is a factor of \(\mathrm { p } ( x )\)
Use the factor theorem to show \(a = 2\)
6
(ii) State the transformation which maps the curve with equation
$$y = x ^ { 2 } ( 2 x + 2 )$$
onto the curve with equation
$$y = x ^ { 2 } ( 2 x + 2 ) + 36$$
6
(iii) The polynomial \(x ^ { 2 } ( 2 x + 2 ) + 36\) can be written as \(( x + 3 ) \left( 2 x ^ { 2 } + b x + c \right)\)
Without finding the values of \(b\) and \(c\), use your answers to parts (a) and (b)(ii) to explain why
$$b ^ { 2 } < 8 c$$