1.02j Manipulate polynomials: expanding, factorising, division, factor theorem

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 \(x ^ { 4 } + 3 x ^ { 3 } + a x + b\), 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 \(b\).

2 The polynomial \(6 x ^ { 3 } + a x ^ { 2 } + b x - 2\), where \(a\) and \(b\) are constants, is denoted by \(\mathrm { p } ( x )\). It is given that \(( 2 x + 1 )\) is a factor of \(\mathrm { p } ( x )\) and that when \(\mathrm { p } ( x )\) is divided by \(( x + 2 )\) the remainder is - 24 . Find the values of \(a\) and \(b\).

6 The polynomial \(8 x ^ { 3 } + a x ^ { 2 } + b x - 1\), where \(a\) and \(b\) are constants, is denoted by \(\mathrm { p } ( x )\). It is given that \(( x + 1 )\) is a factor of \(\mathrm { p } ( x )\) and that when \(\mathrm { p } ( x )\) is divided by ( \(2 x + 1\) ) the remainder is 1 .

1. Find the values of \(a\) and \(b\).
2. When \(a\) and \(b\) have these values, factorise \(\mathrm { p } ( x )\) completely.

2 The cubic equation \(2 x ^ { 3 } - 4 x ^ { 2 } + 3 = 0\) has roots \(\alpha , \beta , \gamma\). Let \(\mathrm { S } _ { \mathrm { n } } = \alpha ^ { \mathrm { n } } + \beta ^ { \mathrm { n } } + \gamma ^ { \mathrm { n } }\).

1. State the value of \(S _ { 1 }\) and find the value of \(S _ { 2 }\).
2. 1. Express \(\mathrm { S } _ { \mathrm { n } + 3 }\) in terms of \(\mathrm { S } _ { \mathrm { n } + 2 }\) and \(\mathrm { S } _ { \mathrm { n } }\).
   2. Hence, or otherwise, find the value of \(S _ { 4 }\).
3. Use the substitution \(\mathrm { y } = \mathrm { S } _ { 1 } - \mathrm { x }\), where \(S _ { 1 }\) is the numerical value found in part (a), to find and simplify an equation whose roots are \(\alpha + \beta , \beta + \gamma , \gamma + \alpha\).
4. Find the value of \(\frac { 1 } { \alpha + \beta } + \frac { 1 } { \beta + \gamma } + \frac { 1 } { \gamma + \alpha }\).

5 The polynomial \(\mathrm { p } ( x )\) is defined by $$\mathrm { p } ( x ) = 5 x ^ { 3 } + a x ^ { 2 } + b x - 16$$ where \(a\) and \(b\) are constants. It is given that \(( x - 2 )\) is a factor of \(\mathrm { p } ( x )\) and that the remainder is 27 when \(\mathrm { p } ( x )\) is divided by \(( x + 1 )\).

1. Find the values of \(a\) and \(b\).
2. Hence factorise \(\mathrm { p } ( x )\) completely.

1 The polynomial \(\mathrm { p } ( x )\) is defined by $$\mathrm { p } ( x ) = 4 x ^ { 3 } + ( k + 1 ) x ^ { 2 } - m x + 3 k$$ where \(k\) and \(m\) are constants. Given that \(( x + 1 )\) is a factor of \(\mathrm { p } ( x )\), express \(m\) in terms of \(k\).

1 Find the quotient and the remainder when \(2 x ^ { 3 } + 3 x ^ { 2 } + 10\) is divided by \(( x + 2 )\).

6 The polynomial \(\mathrm { p } ( x )\) is defined by $$\mathrm { p } ( x ) = a x ^ { 3 } + b x ^ { 2 } - 17 x - a$$ where \(a\) and \(b\) are constants. It is given that \(( x + 2 )\) is a factor of \(\mathrm { p } ( x )\) and that the remainder is 28 when \(\mathrm { p } ( x )\) is divided by \(( x - 2 )\).

1. Find the values of \(a\) and \(b\).
2. Hence factorise \(\mathrm { p } ( x )\) completely.
3. State the number of roots of the equation \(\mathrm { p } \left( 2 ^ { y } \right) = 0\), justifying your answer. \includegraphics[max width=\textwidth, alt={}, center]{17025451-6f07-4f35-9dfc-869e084b5ed0-10_508_538_310_799} The diagram shows part of the curve $$y = 2 \cos 2 x \cos \left( 2 x + \frac { 1 } { 6 } \pi \right)$$ The shaded region is bounded by the curve and the two axes.
4. Show that \(2 \cos 2 x \cos \left( 2 x + \frac { 1 } { 6 } \pi \right)\) can be expressed in the form $$k _ { 1 } ( 1 + \cos 4 x ) + k _ { 2 } \sin 4 x ,$$ where the values of the constants \(k _ { 1 }\) and \(k _ { 2 }\) are to be determined.
5. Find the exact area of the shaded region.

4

1. Find the quotient when \(4 x ^ { 3 } + 8 x ^ { 2 } + 11 x + 9\) is divided by ( \(2 x + 1\) ), and show that the remainder is 5 .
2. Show that the equation \(4 x ^ { 3 } + 8 x ^ { 2 } + 11 x + 4 = 0\) has exactly one real root.

2 The cubic polynomial \(2 x ^ { 3 } + a x ^ { 2 } + b\) is denoted by \(\mathrm { f } ( x )\). It is given that ( \(x + 1\) ) is a factor of \(\mathrm { f } ( x )\), and that when \(\mathrm { f } ( x )\) is divided by \(( x + 2 )\) the remainder is - 5 . Find the values of \(a\) and \(b\).

4 The cubic polynomial \(2 x ^ { 3 } - 5 x ^ { 2 } + a x + b\) is denoted by \(\mathrm { f } ( x )\). It is given that ( \(x - 2\) ) is a factor of \(\mathrm { f } ( x )\), and that when \(\mathrm { f } ( x )\) is divided by \(( x + 1 )\) the remainder is - 6 . Find the values of \(a\) and \(b\).

2 The polynomial \(x ^ { 3 } + 2 x ^ { 2 } + 2 x + 3\) is denoted by \(\mathrm { p } ( x )\).

1. Find the remainder when \(\mathrm { p } ( x )\) is divided by \(x - 1\).
2. Find the quotient and remainder when \(\mathrm { p } ( x )\) is divided by \(x ^ { 2 } + x - 1\).

3 The polynomial \(4 x ^ { 3 } - 7 x + a\), where \(a\) is a constant, is denoted by \(\mathrm { p } ( x )\). It is given that ( \(2 x - 3\) ) is a factor of \(\mathrm { p } ( x )\).

1. Show that \(a = - 3\).
2. Hence, or otherwise, solve the equation \(\mathrm { p } ( x ) = 0\).

5 The polynomial \(3 x ^ { 3 } + 8 x ^ { 2 } + a x - 2\), where \(a\) is a constant, is denoted by \(\mathrm { p } ( x )\). It is given that \(( x + 2 )\) is a factor of \(\mathrm { p } ( x )\).

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

2 The polynomial \(2 x ^ { 3 } - x ^ { 2 } + a x - 6\), where \(a\) is a constant, is denoted by \(\mathrm { p } ( x )\). It is given that ( \(x + 2\) ) is a factor of \(\mathrm { p } ( x )\).

1. Find the value of \(a\).
2. When \(a\) has this value, factorise \(\mathrm { p } ( x )\) completely.

3 The polynomial \(4 x ^ { 3 } - 8 x ^ { 2 } + a x - 3\), where \(a\) is a constant, is denoted by \(\mathrm { p } ( x )\). It is given that ( \(2 x + 1\) ) is a factor of \(\mathrm { p } ( x )\).

1. Find the value of \(a\).
2. When \(a\) has this value, factorise \(\mathrm { p } ( x )\) completely.

5 The polynomial \(a x ^ { 3 } + b x ^ { 2 } - 5 x + 2\), where \(a\) and \(b\) are constants, is denoted by \(\mathrm { p } ( x )\). It is given that \(( x + 1 )\) and \(( x - 2 )\) are factors of \(\mathrm { p } ( x )\).

1. Find the values of \(a\) and \(b\).
2. When \(a\) and \(b\) have these values, find the other linear factor of \(\mathrm { p } ( x )\).

7 The polynomial \(3 x ^ { 3 } + 2 x ^ { 2 } + a x + b\), where \(a\) and \(b\) are constants, is denoted by \(\mathrm { p } ( x )\). It is given that \(( x - 1 )\) is a factor of \(\mathrm { p } ( x )\), and that when \(\mathrm { p } ( x )\) is divided by \(( x - 2 )\) the remainder is 10 .

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

3 The polynomial \(x ^ { 3 } + 4 x ^ { 2 } + a x + 2\), where \(a\) is a constant, is denoted by \(\mathrm { p } ( x )\). It is given that the remainder when \(\mathrm { p } ( x )\) is divided by ( \(x + 1\) ) is equal to the remainder when \(\mathrm { p } ( x )\) is divided by ( \(x - 2\) ).

1. Find the value of \(a\).
2. When \(a\) has this value, show that \(( x - 1 )\) is a factor of \(\mathrm { p } ( x )\) and find the quotient when \(\mathrm { p } ( x )\) is divided by \(( x - 1 )\).

5 The polynomial \(4 x ^ { 3 } + a x ^ { 2 } + 9 x + 9\), where \(a\) is a constant, is denoted by \(\mathrm { p } ( x )\). It is given that when \(\mathrm { p } ( x )\) is divided by \(( 2 x - 1 )\) the remainder is 10 .

1. Find the value of \(a\) and hence verify that ( \(x - 3\) ) is a factor of \(\mathrm { p } ( x )\).
2. When \(a\) has this value, solve the equation \(\mathrm { p } ( x ) = 0\).

7 The polynomial \(a x ^ { 3 } - 3 x ^ { 2 } - 11 x + b\), where \(a\) and \(b\) are constants, is denoted by \(\mathrm { p } ( x )\). It is given that \(( x + 2 )\) is a factor of \(\mathrm { p } ( x )\), and that when \(\mathrm { p } ( x )\) is divided by \(( x + 1 )\) the remainder is 12 .

1. Find the values of \(a\) and \(b\).
2. When \(a\) and \(b\) have these values, factorise \(\mathrm { p } ( x )\) completely.

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 ( \(x ^ { 2 } - 2\) ).

3 The polynomial \(x ^ { 4 } - 4 x ^ { 3 } + 3 x ^ { 2 } + 4 x - 4\) is denoted by \(\mathrm { p } ( x )\).

1. Find the quotient when \(\mathrm { p } ( x )\) is divided by \(x ^ { 2 } - 3 x + 2\).
2. Hence solve the equation \(\mathrm { p } ( x ) = 0\).