Find constants with divisibility condition

3 The polynomial \(x ^ { 4 } + 4 x ^ { 2 } + x + a\) is denoted by \(\mathrm { p } ( x )\). It is given that ( \(x ^ { 2 } + x + 2\) ) is a factor of \(\mathrm { p } ( x )\).
Find the value of \(a\) and the other quadratic factor of \(p ( x )\).

4 The polynomial \(x ^ { 4 } - 2 x ^ { 3 } - 2 x ^ { 2 } + a\) is denoted by \(\mathrm { f } ( x )\). It is given that \(\mathrm { f } ( x )\) is divisible by \(x ^ { 2 } - 4 x + 4\).

1. Find the value of \(a\).
2. When \(a\) has this value, show that \(\mathrm { f } ( x )\) is never negative.

5 The polynomial \(x ^ { 4 } + 5 x + a\) is denoted by \(\mathrm { p } ( x )\). It is given that \(x ^ { 2 } - x + 3\) is a factor of \(\mathrm { p } ( x )\).

1. Find the value of \(a\) and factorise \(\mathrm { p } ( x )\) completely.
2. Hence state the number of real roots of the equation \(\mathrm { p } ( x ) = 0\), justifying your answer.

2 The polynomial \(x ^ { 4 } + 3 x ^ { 2 } + a\), where \(a\) is a constant, is denoted by \(\mathrm { p } ( x )\). It is given that \(x ^ { 2 } + x + 2\) is a factor of \(\mathrm { p } ( x )\). Find the value of \(a\) and the other quadratic factor of \(\mathrm { p } ( x )\).

5 The polynomial \(4 x ^ { 3 } - 4 x ^ { 2 } + 3 x + a\), where \(a\) is a constant, is denoted by \(\mathrm { p } ( x )\). It is given that \(\mathrm { p } ( x )\) is divisible by \(2 x ^ { 2 } - 3 x + 3\).

1. Find the value of \(a\).
2. When \(a\) has this value, solve the inequality \(\mathrm { p } ( x ) < 0\), justifying your answer.

3 The polynomial \(x ^ { 4 } + 3 x ^ { 3 } + a x + 3\) is denoted by \(\mathrm { p } ( x )\). It is given that \(\mathrm { p } ( x )\) is divisible by \(x ^ { 2 } - x + 1\).

1. Find the value of \(a\).
2. When \(a\) has this value, find the real roots of the equation \(\mathrm { p } ( x ) = 0\).

4 The polynomial \(4 x ^ { 4 } + a x ^ { 2 } + 11 x + b\), where \(a\) and \(b\) are constants, is denoted by \(\mathrm { p } ( x )\). It is given that \(\mathrm { p } ( x )\) is divisible by \(x ^ { 2 } - x + 2\).

1. Find the values of \(a\) and \(b\).
2. When \(a\) and \(b\) have these values, find the real roots of the equation \(\mathrm { p } ( x ) = 0\).

3 The polynomial \(2 x ^ { 4 } + a x ^ { 3 } + b x - 1\), where \(a\) and \(b\) are constants, is denoted by \(\mathrm { p } ( x )\). When \(\mathrm { p } ( x )\) is divided by \(x ^ { 2 } - x + 1\) the remainder is \(3 x + 2\). Find the values of \(a\) and \(b\).

3. The diagram shows the curve with equation \(y = x ^ { 3 } + a x ^ { 2 } + b x + c\), where \(a , b\) and \(c\) are constants. The curve crosses the \(x\)-axis at the point \(( - 1,0 )\) and touches the \(x\)-axis at the point \(( 3,0 )\). Show that \(a = - 5\) and find the values of \(b\) and \(c\).