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

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1. Give the complete expansion of \(\left( x + \frac { 2 } { x } \right) ^ { 5 }\).
2. In the expansion of \(\left( a + b x ^ { 2 } \right) \left( x + \frac { 2 } { x } \right) ^ { 5 }\), the coefficient of \(x\) is zero and the coefficient of \(\frac { 1 } { x }\) is 80 . Find the values of the constants \(a\) and \(b\).

3 Solve, by factorising, the equation $$6 \cos \theta \tan \theta - 3 \cos \theta + 4 \tan \theta - 2 = 0$$ for \(0 ^ { \circ } \leqslant \theta \leqslant 180 ^ { \circ }\).

4

1. Find the first three terms in the expansion of \(( 2 + a x ) ^ { 5 }\) in ascending powers of \(x\).
2. Given that the coefficient of \(x ^ { 2 }\) in the expansion of \(( 1 + 2 x ) ( 2 + a x ) ^ { 5 }\) is 240 , find the possible values of \(a\).

2 The polynomial \(\mathrm { p } ( x )\) is defined by $$\mathrm { p } ( x ) = 6 x ^ { 3 } + a x ^ { 2 } + 9 x + b$$ where \(a\) and \(b\) are constants. It is given that \(( x - 2 )\) and \(( 2 x + 1 )\) are factors of \(\mathrm { p } ( x )\).
Find the values of \(a\) and \(b\).

6 The polynomial \(\mathrm { p } ( x )\) is defined by $$\mathrm { p } ( x ) = 6 x ^ { 3 } + a x ^ { 2 } - 4 x - 3$$ where \(a\) is a constant. It is given that \(( x + 3 )\) is a factor of \(\mathrm { p } ( x )\).

1. Find the value of \(a\).
2. Using this value of \(a\), factorise \(\mathrm { p } ( x )\) completely.
3. Hence solve the equation \(\mathrm { p } ( \operatorname { cosec } \theta ) = 0\) for \(0 ^ { \circ } < \theta < 360 ^ { \circ }\).

7 The polynomial \(\mathrm { p } ( x )\) is defined by $$\mathrm { p } ( x ) = a x ^ { 3 } - 11 x ^ { 2 } - 19 x - a$$ where \(a\) is a constant. It is given that \(( x - 3 )\) 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. Hence find the exact values of \(y\) that satisfy the equation \(\mathrm { p } \left( \mathrm { e } ^ { y } + \mathrm { e } ^ { - y } \right) = 0\).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

5

1. Find the quotient when \(x ^ { 4 } - 32 x + 55\) is divided by \(( x - 2 ) ^ { 2 }\) and show that the remainder is 7 .
2. Factorise \(x ^ { 4 } - 32 x + 48\).
3. Hence solve the equation \(\mathrm { e } ^ { - 12 y } - 32 \mathrm { e } ^ { - 3 y } + 48 = 0\), giving your answer in an exact form.

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.

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

1. Find the value of \(a\) and hence factorise \(\mathrm { p } ( x )\).
2. Show that the equation \(\mathrm { p } \left( \mathrm { e } ^ { 3 y } \right) = 0\) has only one real root and find its exact value.

4 The polynomial \(\mathrm { p } ( x )\) is defined by $$\mathrm { p } ( x ) = 2 x ^ { 3 } + 3 x ^ { 2 } + k x - 30$$ where \(k\) is a constant. It is given that \(( x - 3 )\) is a factor of \(\mathrm { p } ( x )\).

1. Find the value of \(k\).
2. Hence find the quotient when \(\mathrm { p } ( x )\) is divided by ( \(x - 3\) ) and factorise \(\mathrm { p } ( x )\) completely.
3. It is given that \(a\) is one of the roots of the equation \(\mathrm { p } ( x ) = 0\). Given also that the equation \(| 4 y - 5 | = a\) is satisfied by two real values of \(y\), find these two values of \(y\).

7 A curve has parametric equations $$x = \frac { 2 t + 3 } { t + 2 } , \quad y = t ^ { 2 } + a t + 1$$ where \(a\) is a constant. It is given that, at the point \(P\) on the curve, the gradient is 1 .

1. Show that the value of \(t\) at \(P\) satisfies the equation $$2 t ^ { 3 } + ( a + 8 ) t ^ { 2 } + ( 4 a + 8 ) t + 4 a - 1 = 0$$
2. It is given that \(( t + 1 )\) is a factor of $$2 t ^ { 3 } + ( a + 8 ) t ^ { 2 } + ( 4 a + 8 ) t + 4 a - 1$$ Find the value of \(a\).
3. Hence show that \(P\) is the only point on the curve at which the gradient is 1 .
   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.

5 The polynomial \(\mathrm { p } ( x )\) is defined by \(\mathrm { p } ( x ) = 9 x ^ { 3 } + 18 x ^ { 2 } + 5 x + 4\).

1. Find the quotient when \(\mathrm { p } ( x )\) is divided by \(( 3 x + 2 )\), and show that the remainder is 6 . \includegraphics[max width=\textwidth, alt={}, center]{76df3465-9617-4f2b-a8b7-f474b2817504-08_2713_33_146_2012} \includegraphics[max width=\textwidth, alt={}, center]{76df3465-9617-4f2b-a8b7-f474b2817504-09_2723_33_138_20}
2. Find the value of \(\int _ { 0 } ^ { 2 } \frac { \mathrm { p } ( x ) } { 3 x + 2 } \mathrm {~d} x\) ,giving your answer in the form \(a + \ln b\) where \(a\) and \(b\) are integers.

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1. Find the quotient when \(4 x ^ { 3 } + 17 x ^ { 2 } + 9 x\) is divided by \(x ^ { 2 } + 5 x + 6\), and show that the remainder is 18 .
2. Hence solve the equation \(4 x ^ { 3 } + 17 x ^ { 2 } + 9 x - 18 = 0\).

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

1. Find the values of \(a\) and \(b\).
2. Hence find the exact root of the equation \(\mathrm { p } \left( \mathrm { e } ^ { 2 y } \right) = 0\).

6 The polynomial \(\mathrm { p } ( x )\) is defined by $$\mathrm { p } ( x ) = 4 x ^ { 3 } + 16 x ^ { 2 } + 9 x - 15$$

1. Find the quotient when \(\mathrm { p } ( x )\) is divided by \(( 2 x + 3 )\), and show that the remainder is - 6 .
2. Find \(\int \frac { \mathrm { p } ( x ) } { 2 x + 3 } \mathrm {~d} x\).
3. Factorise \(\mathrm { p } ( x ) + 6\) completely and hence solve the equation $$p ( \operatorname { cosec } 2 \theta ) + 6 = 0$$ for \(0 ^ { \circ } < \theta < 135 ^ { \circ }\).

2 The polynomial \(\mathrm { p } ( x )\) is defined by $$\mathrm { p } ( x ) = x ^ { 3 } + a x ^ { 2 } + b x + 16$$ where \(a\) and \(b\) are constants. It is given that \(( x + 2 )\) is a factor of \(\mathrm { p } ( x )\) and that the remainder is 72 when \(\mathrm { p } ( x )\) is divided by \(( x - 2 )\). Find the values of \(a\) and \(b\).

7 \includegraphics[max width=\textwidth, alt={}, center]{c473f577-1e96-4d11-a0d5-cdfa4873c295-12_650_720_260_708} A curve has equation \(y = \mathrm { f } ( x )\) where \(\mathrm { f } ( x ) = x ^ { 4 } - 5 x ^ { 3 } + 6 x ^ { 2 } + 5 x - 15\). As shown in the diagram, the curve crosses the \(x\)-axis at the points \(A\) and \(B\) with coordinates \(( a , 0 )\) and \(( b , 0 )\) respectively.

1. Use the factor theorem to show that \(( x - 3 )\) is a factor of \(\mathrm { f } ( x )\).
2. By first finding the quotient when \(\mathrm { f } ( x )\) is divided by \(( x - 3 )\), show that $$a = - \sqrt { \frac { 5 } { 2 - a } } .$$
3. Use an iterative formula, based on the equation in part (b), to find the value of \(a\) correct to 3 significant figures. Give the result of each iteration to 5 significant figures.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

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 \(0 < \theta < 2 \pi\).

1 The polynomial \(\mathrm { p } ( x )\) is defined by $$\mathrm { p } ( x ) = a x ^ { 3 } + b x - 10$$ where \(a\) and \(b\) are constants. It is given that \(( x + 2 )\) is a factor of \(\mathrm { p } ( x )\) and that the remainder is - 55 when \(\mathrm { p } ( x )\) is divided by \(( x + 3 )\).

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

6 The polynomial \(\mathrm { p } ( x )\) is defined by $$\mathrm { p } ( x ) = 12 x ^ { 3 } - 9 x ^ { 2 } + 8 x - 4$$

1. Find the quotient when \(\mathrm { p } ( x )\) is divided by \(( 4 x - 3 )\) and show that the remainder is 2 .
2. Hence find \(\int _ { 2 } ^ { 12 } \left( \frac { \mathrm { p } ( x ) } { 4 x - 3 } - 3 x ^ { 2 } \right) \mathrm { d } x\), giving your answer in the form \(a + \ln b\).

4 The polynomial \(\mathrm { p } ( x )\) is defined by $$\mathrm { p } ( x ) = a x ^ { 3 } + 23 x ^ { 2 } - a x - 8$$ where \(a\) is a constant. It is given that \(( 2 x + 1 )\) is a factor of \(\mathrm { p } ( x )\).

1. Find the value of \(a\) and hence factorise \(\mathrm { p } ( x )\) completely.
2. Hence solve the equation \(\mathrm { p } \left( \mathrm { e } ^ { 4 y } \right) = 0\), giving your answer correct to 3 significant figures.

5 The polynomial \(\mathrm { p } ( x )\) is defined by $$\mathrm { p } ( x ) = 6 x ^ { 3 } + a x ^ { 2 } + b x - 20$$ where \(a\) and \(b\) are constants. It is given that \(( x + 2 )\) is a factor of \(\mathrm { p } ( x )\) and that the remainder is - 11 when \(\mathrm { p } ( x )\) is divided by \(( x + 1 )\).

1. Find the values of \(a\) and \(b\).
2. Hence factorise \(\mathrm { p } ( x )\), and determine the exact roots of the equation \(\mathrm { p } ( 3 x ) = 0\).

1 When the polynomial $$a x ^ { 3 } + 4 a x ^ { 2 } - 7 x - 5$$ is divided by \(( x + 2 )\), the remainder is 33 .
Find the value of the constant \(a\).

4 The polynomial \(\mathrm { p } ( x )\) is defined by $$\mathrm { p } ( x ) = a x ^ { 3 } - a x ^ { 2 } - 15 x + 18$$ where \(a\) is a constant. It is given that ( \(x + 2\) ) is a factor of \(\mathrm { p } ( x )\).

1. Find the value of \(a\).
2. Hence factorise \(\mathrm { p } ( x )\) completely.
   
3. Solve the equation \(\mathrm { p } \left( \operatorname { cosec } ^ { 2 } \theta \right) = 0\) for \(- 90 ^ { \circ } < \theta < 90 ^ { \circ }\).