7 In this question you must show detailed reasoning.
- Nigel is asked to determine whether \(( x + 7 )\) is a factor of \(x ^ { 3 } - 37 x + 84\). He substitutes \(x = 7\) and calculates \(7 ^ { 3 } - 37 \times 7 + 84\). This comes to 168 , so Nigel concludes that ( \(x + 7\) ) is not a factor.
Nigel's conclusion is wrong.
- Explain why Nigel's argument is not valid.
- Show that \(( x + 7 )\) is a factor of \(x ^ { 3 } - 37 x + 84\).
- Sketch the graph of \(y = x ^ { 3 } - 37 x + 84\), indicating the coordinates of the points at which the curve crosses the coordinate axes.
- The graph in part (b) is translated by \(\binom { 1 } { 0 }\). Find the equation of the translated graph, giving your answer in the form \(y = x ^ { 3 } + a x ^ { 2 } + b x + c\) where \(a , b\) and \(c\) are integers.