Multiple unknowns with derivative condition

5 The polynomial \(2 x ^ { 3 } + a x ^ { 2 } + b x - 4\), where \(a\) and \(b\) are constants, is denoted by \(\mathrm { p } ( x )\). The result of differentiating \(\mathrm { p } ( x )\) with respect to \(x\) is denoted by \(\mathrm { p } ^ { \prime } ( x )\). It is given that \(( x + 2 )\) is a factor of \(\mathrm { p } ( x )\) and of \(\mathrm { p } ^ { \prime } ( x )\).

1. Find the values of \(a\) and \(b\).
2. When \(a\) and \(b\) have these values, factorise \(\mathrm { p } ( x )\) completely.

5 The polynomial \(a x ^ { 3 } - 10 x ^ { 2 } + b x + 8\), where \(a\) and \(b\) are constants, is denoted by \(\mathrm { p } ( x )\). It is given that \(( x - 2 )\) is a factor of both \(\mathrm { p } ( x )\) and \(\mathrm { p } ^ { \prime } ( x )\).

1. Find the values of \(a\) and \(b\).
2. When \(a\) and \(b\) have these values, factorise \(\mathrm { p } ( x )\) completely.

7. A curve \(C\) has equation \(y = \mathrm { f } ( x )\). Given that

• \(\mathrm { f } ^ { \prime } ( x ) = 18 x ^ { 2 } + 2 a x + b\)
• the \(y\)-intercept of \(C\) is - 48
• the point \(A\), with coordinates \(( - 1,45 )\) lies on \(C\)
  a. Show that \(a - b = 99\)
  b. Find the value of \(a\) and the value of \(b\).
  c. Show that \(( 2 x + 1 )\) is a factor of \(\mathrm { f } ( x )\).

3. $$\mathrm { g } ( x ) = 4 x ^ { 3 } + a x ^ { 2 } + 4 x + b$$ where \(a\) and \(b\) are constants.
Given that

• ( \(2 x + 1\) ) is a factor of \(\mathrm { g } ( x )\)
• the curve with equation \(y = \mathrm { g } ( x )\) has a point of inflection at \(x = \frac { 1 } { 6 }\)
  1. find the value of \(a\) and the value of \(b\)
  2. Show that there are no stationary points on the curve with equation \(y = \mathrm { g } ( x )\).