Polynomial Identity Matching

2 The polynomial \(x ^ { 4 } - 9 x ^ { 2 } - 6 x - 1\) is denoted by \(\mathrm { f } ( x )\).

1. Find the value of the constant \(a\) for which $$f ( x ) \equiv \left( x ^ { 2 } + a x + 1 \right) \left( x ^ { 2 } - a x - 1 \right)$$
2. Hence solve the equation \(\mathrm { f } ( x ) = 0\), giving your answers in an exact form.

1. Given that

$$( x - 5 ) ( 2 x + 1 ) ( x + 3 ) \equiv a x ^ { 3 } + b x ^ { 2 } - 32 x - 15$$ where \(a\) and \(b\) are constants,

1. find the value of \(a\) and the value of \(b\).
2. Hence find $$\int \frac { ( x - 5 ) ( 2 x + 1 ) ( x + 3 ) } { 5 \sqrt { x } } \mathrm {~d} x$$ writing each term in simplest form.

1. Given that

$$\left( x ^ { 2 } + 2 x - 3 \right) \left( 2 x ^ { 2 } + k x + 7 \right) \equiv 2 x ^ { 4 } + A x ^ { 3 } + A x ^ { 2 } + B x - 21 ,$$ find the values of the constants \(k , A\) and \(B\).

4 Find the values of \(A , B , C\) and \(D\) in the identity \(3 x ^ { 3 } - x ^ { 2 } + 2 \equiv A ( x - 1 ) ^ { 3 } + \left( x ^ { 3 } + B x ^ { 2 } + C x + D \right)\).

2 Given that $$( x - p ) \left( 2 x ^ { 2 } + 9 x + 10 \right) = \left( x ^ { 2 } - 4 \right) ( 2 x + q )$$ for all values of \(x\), find the constants \(p\) and \(q\).

6 Show that the expression \(3 x ^ { 3 } + x ^ { 2 } - 6 x - 5\) can be written in the form \(( x + 2 ) \left( a x ^ { 2 } + b x + c \right) + d\) where \(a\), \(b\), \(c\) and \(d\) are constants to be determined.

Find the values of \(A\), \(B\), \(C\) and \(D\) in the identity $$2x^3 - 3x^2 + x - 2 \equiv (x + 2)(Ax^2 + Bx + C) + D.$$ [5]