Finding Constants from Division Conditions

7 The polynomial \(\mathrm { p } ( x )\) is defined by $$\mathrm { p } ( x ) = 2 x ^ { 3 } + 5 x ^ { 2 } + a x + 2 a$$ where \(a\) is an integer.

1. Find, in terms of \(x\) and \(a\), the quotient when \(\mathrm { p } ( x )\) is divided by ( \(x + 2\) ), and show that the remainder is 4 .
2. It is given that \(\int _ { - 1 } ^ { 1 } \frac { \mathrm { p } ( x ) } { x + 2 } \mathrm {~d} x = \frac { 22 } { 3 } + \ln b\), where \(b\) is an integer. Find the values of \(a\) and \(b\).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

7 The polynomial \(\mathrm { p } ( x )\) is defined by $$p ( x ) = 9 x ^ { 3 } + 6 x ^ { 2 } + 12 x + k$$ where \(k\) is a constant.

1. Find the quotient when \(\mathrm { p } ( x )\) is divided by \(( 3 x + 2 )\) and show that the remainder is \(( k - 8 )\).
2. It is given that \(\int _ { 1 } ^ { 6 } \frac { \mathrm { p } ( \mathrm { x } ) } { 3 \mathrm { x } + 2 } \mathrm { dx } = \mathrm { a } + \ln 64\), where \(a\) is an integer. Find the values of \(a\) and \(k\).
   If you use the following page to complete the answer to any question, the question number must be clearly shown.

4 The polynomial \(\mathrm { f } ( x )\) is defined by $$f ( x ) = 3 x ^ { 3 } + a x ^ { 2 } + a x + a$$ where \(a\) is a constant.

1. Given that \(( x + 2 )\) is a factor of \(\mathrm { f } ( x )\), find the value of \(a\).
2. When \(a\) has the value found in part (i), find the quotient when \(\mathrm { f } ( x )\) is divided by ( \(x + 2\) ).

2

1. Find the quotient and remainder when \(2 x ^ { 3 } - 7 x ^ { 2 } - 9 x + 3\) is divided by \(x ^ { 2 } - 2 x + 5\).
2. Hence find the values of the constants \(p\) and \(q\) such that \(x ^ { 2 } - 2 x + 5\) is a factor of \(2 x ^ { 3 } - 7 x ^ { 2 } + p x + q\).

3 The polynomial \(x ^ { 4 } + 3 x ^ { 3 } + a x + b\), 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 \(2 x + 3\). Find the values of \(a\) and \(b\).

6

1. The polynomial \(x ^ { 4 } + a x ^ { 3 } - x ^ { 2 } + b x + 2\), where \(a\) and \(b\) are constants, is denoted by \(\mathrm { p } ( x )\). It is given that \(( x - 1 )\) and \(( x + 2 )\) are factors of \(\mathrm { p } ( x )\). Find the values of \(a\) and \(b\).
2. When \(a\) and \(b\) have these values, find the quotient when \(\mathrm { p } ( x )\) is divided by \(x ^ { 2 } + x - 2\).

7 The polynomial \(2 x ^ { 3 } - 4 x ^ { 2 } + a x + b\), where \(a\) and \(b\) are constants, is denoted by \(\mathrm { p } ( x )\). It is given that when \(\mathrm { p } ( x )\) is divided by \(( x + 1 )\) the remainder is 4 , and that when \(\mathrm { p } ( x )\) is divided by \(( x - 3 )\) the remainder is 12 .

1. Find the values of \(a\) and \(b\).
2. When \(a\) and \(b\) have these values, find the quotient and remainder when \(\mathrm { p } ( x )\) is divided by ( \(x ^ { 2 } - 2\) ).

4

1. The polynomial \(x ^ { 3 } + a x ^ { 2 } + b x + 8\), where \(a\) and \(b\) are constants, is denoted by \(\mathrm { p } ( x )\). It is given that when \(\mathrm { p } ( x )\) is divided by \(( x - 3 )\) the remainder is 14 , and that when \(\mathrm { p } ( x )\) is divided by \(( x + 2 )\) the remainder is 24 . Find the values of \(a\) and \(b\).
2. When \(a\) and \(b\) have these values, find the quotient when \(\mathrm { p } ( x )\) is divided by \(x ^ { 2 } + 2 x - 8\) and hence solve the equation \(\mathrm { p } ( x ) = 0\).

6

1. Find the quotient and remainder when $$x ^ { 4 } + x ^ { 3 } + 3 x ^ { 2 } + 12 x + 6$$ is divided by ( \(x ^ { 2 } - x + 4\) ).
2. It is given that, when $$x ^ { 4 } + x ^ { 3 } + 3 x ^ { 2 } + p x + q$$ is divided by ( \(x ^ { 2 } - x + 4\) ), the remainder is zero. Find the values of the constants \(p\) and \(q\).
3. When \(p\) and \(q\) have these values, show that there is exactly one real value of \(x\) satisfying the equation $$x ^ { 4 } + x ^ { 3 } + 3 x ^ { 2 } + p x + q = 0$$ and state what that value is.

6

1. Find the quotient and remainder when $$x ^ { 4 } + x ^ { 3 } + 3 x ^ { 2 } + 12 x + 6$$ is divided by ( \(x ^ { 2 } - x + 4\) ).
2. It is given that, when $$x ^ { 4 } + x ^ { 3 } + 3 x ^ { 2 } + p x + q$$ is divided by \(\left( x ^ { 2 } - x + 4 \right)\), the remainder is zero. Find the values of the constants \(p\) and \(q\).
3. When \(p\) and \(q\) have these values, show that there is exactly one real value of \(x\) satisfying the equation $$x ^ { 4 } + x ^ { 3 } + 3 x ^ { 2 } + p x + q = 0$$ and state what that value is.

3

1. Find the quotient and the remainder when \(3 x ^ { 3 } - 2 x ^ { 2 } + x + 7\) is divided by \(x ^ { 2 } - 2 x + 5\).
2. Hence, or otherwise, determine the values of the constants \(a\) and \(b\) such that, when \(3 x ^ { 3 } - 2 x ^ { 2 } + a x + b\) is divided by \(x ^ { 2 } - 2 x + 5\), there is no remainder.

3 When \(x ^ { 4 } - 2 x ^ { 3 } - 7 x ^ { 2 } + 7 x + a\) is divided by \(x ^ { 2 } + 2 x - 1\), the quotient is \(x ^ { 2 } + b x + 2\) and the remainder is \(c x + 7\). Find the values of the constants \(a , b\) and \(c\).