Two polynomials, shared factor or separate conditions

6 The polynomials \(\mathrm { f } ( x )\) and \(\mathrm { g } ( x )\) are defined by $$\mathrm { f } ( x ) = 4 x ^ { 3 } + a x ^ { 2 } + 8 x + 15 \quad \text { and } \quad \mathrm { g } ( x ) = x ^ { 2 } + b x + 18$$ where \(a\) and \(b\) are constants.

1. Given that \(( x + 3 )\) is a factor of \(\mathrm { f } ( x )\), find the value of \(a\).
2. Given that the remainder is 40 when \(\mathrm { g } ( x )\) is divided by \(( x - 2 )\), find the value of \(b\).
3. When \(a\) and \(b\) have these values, factorise \(\mathrm { f } ( x ) - \mathrm { g } ( x )\) completely.
4. Hence solve the equation \(\mathrm { f } ( \operatorname { cosec } \theta ) - \mathrm { g } ( \operatorname { cosec } \theta ) = 0\) for \(0 < \theta < 2 \pi\).

4 The polynomials \(\mathrm { f } ( x )\) and \(\mathrm { g } ( x )\) are defined by $$\mathrm { f } ( x ) = x ^ { 3 } + a x ^ { 2 } + b \quad \text { and } \quad \mathrm { g } ( x ) = x ^ { 3 } + b x ^ { 2 } - a$$ where \(a\) and \(b\) are constants. It is given that ( \(x + 2\) ) is a factor of \(\mathrm { f } ( x )\). It is also given that, when \(\mathrm { g } ( x )\) is divided by \(( x + 1 )\), the remainder is - 18 .

1. Find the values of \(a\) and \(b\).
2. When \(a\) and \(b\) have these values, find the greatest possible value of \(\mathrm { g } ( x ) - \mathrm { f } ( x )\) as \(x\) varies.

4 The polynomials \(\mathrm { p } ( x )\) and \(\mathrm { q } ( x )\) are defined by $$\mathrm { p } ( x ) = x ^ { 3 } + x ^ { 2 } + a x - 15 \quad \text { and } \quad \mathrm { q } ( x ) = 2 x ^ { 3 } + x ^ { 2 } + b x + 21 ,$$ where \(a\) and \(b\) are constants. It is given that \(( x + 3 )\) is a factor of \(\mathrm { p } ( x )\) and also of \(\mathrm { q } ( x )\).

1. Find the values of \(a\) and \(b\).
2. Show that the equation \(\mathrm { q } ( x ) - \mathrm { p } ( x ) = 0\) has only one real root. \includegraphics[max width=\textwidth, alt={}, center]{e2b16207-2cb7-412b-ba7f-758e4d3f1ffb-06_631_643_260_749} The diagram shows the curve \(y = 4 e ^ { - 2 x }\) and a straight line. The curve crosses the \(y\)-axis at the point \(P\). The straight line crosses the \(y\)-axis at the point \(( 0,9 )\) and its gradient is equal to the gradient of the curve at \(P\). The straight line meets the curve at two points, one of which is \(Q\) as shown.

2 The polynomial \(\mathrm { f } ( x )\) is defined by \(\mathrm { f } ( x ) = 2 x ^ { 3 } - 7 x ^ { 2 } + 13\).

1. Use the Remainder Theorem to find the remainder when \(\mathrm { f } ( x )\) is divided by \(( 2 x - 3 )\).
2. The polynomial \(\mathrm { g } ( x )\) is defined by \(\mathrm { g } ( x ) = 2 x ^ { 3 } - 7 x ^ { 2 } + 13 + d\), where \(d\) is a constant. Given that ( \(2 x - 3\) ) is a factor of \(\mathrm { g } ( x )\), show that \(d = - 4\).
3. Express \(\mathrm { g } ( x )\) in the form \(( 2 x - 3 ) \left( x ^ { 2 } + a x + b \right)\).

\(x^2 + bx + c\) and \(x^2 + dx + e\) have a common factor \((x + 2)\) Show that \(2(d - b) = e - c\) Fully justify your answer. [4 marks]