Find constants from coefficient conditions

8 You are given that

• the coefficient of \(x ^ { 3 }\) in the expansion of \(\left( 5 + 2 x ^ { 2 } \right) \left( x ^ { 3 } + k x + m \right)\) is 29 ,
• when \(x ^ { 3 } + k x + m\) is divided by ( \(x - 3\) ), the remainder is 59 .

Find the values of \(k\) and \(m\).

6 You are given that

• the coefficient of \(x ^ { 3 }\) in the expansion of \(\left( 5 + 2 x ^ { 2 } \right) \left( x ^ { 3 } + k x + m \right)\) is 29 ,
• when \(x ^ { 3 } + k x + m\) is divided by ( \(x - 3\) ), the remainder is 59 .

Find the values of \(k\) and \(m\).

4.
\includegraphics[max width=\textwidth, alt={}, center]{fbe229f8-d390-487d-87fc-90edc50c3325-3_554_988_1153_486} The figure above shows the graph of the curve with equation \(y = f ( x )\). The curve crosses the \(x\) axis at the points \(( - 4,0 ) , ( 2,0 )\) and \(( 4,0 )\), and the \(y\) axis at the point \(( 0,16 )\). Determine the equation of \(f ( x )\) in the form $$f ( x ) \equiv a x ^ { 3 } + b x ^ { 2 } + c x + d$$ where \(a , b , c\) and \(d\) are constants.

1 Part of the graph of \(y = \mathrm { f } ( x )\) is shown below, where \(\mathrm { f } ( x )\) is a cubic polynomial.
\includegraphics[max width=\textwidth, alt={}, center]{6a6316e4-7b2d-4533-988a-4863d79ce668-04_681_679_475_694}

1. Find \(\mathrm { f } ( - 1 )\).
2. Write down three linear factors of \(\mathrm { f } ( x )\). It is given that \(\mathrm { f } ( x ) \equiv a x ^ { 3 } + b x ^ { 2 } + c x + d\).
3. Show that \(a = - 2\).
4. Find \(b , c\) and \(d\).