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

1 The polynomial \(\mathrm { f } ( x )\) is defined by \(\mathrm { f } ( x ) = 4 x ^ { 3 } - 13 x + 6\).

1. Find \(\mathrm { f } ( - 2 )\).
2. Use the Factor Theorem to show that \(2 x - 3\) is a factor of \(\mathrm { f } ( x )\).
3. Simplify \(\frac { 2 x ^ { 2 } + x - 6 } { \mathrm { f } ( x ) }\).

5 The polynomial \(\mathrm { f } ( x )\) is defined by \(\mathrm { f } ( x ) = 4 x ^ { 3 } - 11 x - 3\).

1. Use the Factor Theorem to show that ( \(2 x + 3\) ) is a factor of \(\mathrm { f } ( x )\).
2. Write \(\mathrm { f } ( x )\) in the form \(( 2 x + 3 ) \left( a x ^ { 2 } + b x + c \right)\), where \(a , b\) and \(c\) are integers.
3. 1. Show that the equation \(2 \cos 2 \theta \sin \theta + 9 \sin \theta + 3 = 0\) can be written as \(4 x ^ { 3 } - 11 x - 3 = 0\), where \(x = \sin \theta\).
   2. Hence find all solutions of the equation \(2 \cos 2 \theta \sin \theta + 9 \sin \theta + 3 = 0\) in the interval \(0 ^ { \circ } < \theta < 360 ^ { \circ }\), giving your solutions to the nearest degree.

5

1. 1. Express \(3 \sin x + 4 \cos x\) in the form \(R \sin ( x + \alpha )\) where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\), giving your value of \(\alpha\) to the nearest \(0.1 ^ { \circ }\).
   2. Hence solve the equation \(3 \sin 2 \theta + 4 \cos 2 \theta = 5\) in the interval \(0 ^ { \circ } < \theta < 360 ^ { \circ }\), giving your solutions to the nearest \(0.1 ^ { \circ }\).
2. 1. Show that the equation \(\tan 2 \theta \tan \theta = 2\) can be written as \(2 \tan ^ { 2 } \theta = 1\).
   2. Hence solve the equation \(\tan 2 \theta \tan \theta = 2\) in the interval \(0 ^ { \circ } \leqslant \theta \leqslant 180 ^ { \circ }\), giving your solutions to the nearest \(0.1 ^ { \circ }\).
3. 1. Use the Factor Theorem to show that \(2 x - 1\) is a factor of \(8 x ^ { 3 } - 4 x + 1\).
   2. Show that \(4 \cos 2 \theta \cos \theta + 1\) can be written as \(8 x ^ { 3 } - 4 x + 1\) where \(x = \cos \theta\).
   3. Given that \(\theta = 72 ^ { \circ }\) is a solution of \(4 \cos 2 \theta \cos \theta + 1 = 0\), use the results from parts (c)(i) and (c)(ii) to show that the exact value of \(\cos 72 ^ { \circ }\) is \(\frac { ( \sqrt { 5 } - 1 ) } { p }\) where \(p\) is an integer.
      [0pt] [3 marks]

3

1. The polynomial \(\mathrm { f } ( x )\) is defined by \(\mathrm { f } ( x ) = 8 x ^ { 3 } - 12 x ^ { 2 } - 2 x + d\), where \(d\) is a constant. When \(\mathrm { f } ( x )\) is divided by ( \(2 x + 1\) ), the remainder is - 2 . Use the Remainder Theorem to find the value of \(d\).
2. The polynomial \(\mathrm { g } ( x )\) is defined by \(\mathrm { g } ( x ) = 8 x ^ { 3 } - 12 x ^ { 2 } - 2 x + 3\).
   1. Given that \(x = - \frac { 1 } { 2 }\) is a solution of the equation \(\mathrm { g } ( x ) = 0\), write \(\mathrm { g } ( x )\) as a product of three linear factors.
   2. The function h is defined by \(\mathrm { h } ( x ) = \frac { 4 x ^ { 2 } - 1 } { \mathrm {~g} ( x ) }\) for \(x > 2\). Simplify \(\mathrm { h } ( x )\), and hence show that h is a decreasing function.
      [0pt] [4 marks]

2. Given that \(x ^ { 2 } + 4 x + 5\) is a factor of \(x ^ { 3 } + x ^ { 2 } - 7 x - 15\), solve the equation \(x ^ { 3 } + x ^ { 2 } - 7 x - 15 = 0\).

4 In this question you must show detailed reasoning. The cubic polynomial \(6 x ^ { 3 } + k x ^ { 2 } + 57 x - 20\) is denoted by \(\mathrm { f } ( x )\). It is given that \(( 2 x - 1 )\) is a factor of \(\mathrm { f } ( x )\).

1. Use the factor theorem to show that \(k = - 37\).
2. Using this value of \(k\), factorise \(\mathrm { f } ( x )\) completely.
3. 1. Hence find the three values of \(t\) satisfying the equation \(6 \mathrm { e } ^ { - 3 t } - 37 \mathrm { e } ^ { - 2 t } + 57 \mathrm { e } ^ { - t } - 20 = 0\).
   2. Express the sum of the three values found in part (c)(i) as a single logarithm.

8 In this question you must show detailed reasoning. The roots of the equation \(x ^ { 3 } - x ^ { 2 } + k x - 2 = 0\) are \(\alpha , \frac { 1 } { \alpha }\) and \(\beta\).

1. Evaluate, in exact form, the roots of the equation.
2. Find \(k\).

8 In this question you must show detailed reasoning. You are given that i is a root of the equation \(z ^ { 4 } - 2 z ^ { 3 } + 3 z ^ { 2 } + a z + b = 0\), where \(a\) and \(b\) are real constants.

1. Show that \(a = - 2\) and \(b = 2\).
2. Find the other roots of this equation.

4 In this question you must show detailed reasoning. The equation \(z ^ { 3 } + 2 z ^ { 2 } + k z + 3 = 0\), where \(k\) is a constant, has roots \(\alpha , \frac { 1 } { \alpha }\) and \(\beta\).
Determine the roots in exact form.

13 Answer part (i) of this question on the insert provided. The insert shows the graph of \(y = \frac { 1 } { x }\).

1. On the insert, on the same axes, plot the graph of \(y = x ^ { 2 } - 5 x + 5\) for \(0 \leqslant x \leqslant 5\).
2. Show algebraically that the \(x\)-coordinates of the points of intersection of the curves \(y = \frac { 1 } { x }\) and \(y = x ^ { 2 } - 5 x + 5\) satisfy the equation \(x ^ { 3 } - 5 x ^ { 2 } + 5 x - 1 = 0\).
3. Given that \(x = 1\) at one of the points of intersection of the curves, factorise \(x ^ { 3 } - 5 x ^ { 2 } + 5 x - 1\) into a linear and a quadratic factor. Show that only one of the three roots of \(x ^ { 3 } - 5 x ^ { 2 } + 5 x - 1 = 0\) is rational.

7 In this question you must show detailed reasoning. The function \(\mathrm { f } ( x )\) is defined by \(\mathrm { f } ( x ) = x ^ { 3 } + x ^ { 2 } - 8 x - 12\) for all values of \(x\).

1. Use the factor theorem to show that \(( x + 2 )\) is a factor of \(\mathrm { f } ( x )\).
2. Solve the equation \(\mathrm { f } ( x ) = 0\).

1 Part of the graph of \(y = \mathrm { f } ( x )\) is shown below, where \(\mathrm { f } ( x )\) is a cubic polynomial. \includegraphics[max width=\textwidth, alt={}, center]{6a6316e4-7b2d-4533-988a-4863d79ce668-04_681_679_475_694}

1. Find \(\mathrm { f } ( - 1 )\).
2. Write down three linear factors of \(\mathrm { f } ( x )\). It is given that \(\mathrm { f } ( x ) \equiv a x ^ { 3 } + b x ^ { 2 } + c x + d\).
3. Show that \(a = - 2\).
4. Find \(b , c\) and \(d\).

6 In this question you must show detailed reasoning. The cubic polynomial \(\mathrm { f } ( x )\) is defined by \(\mathrm { f } ( x ) = 4 x ^ { 3 } + 4 x ^ { 2 } + 7 x - 5\).

1. Show that \(( 2 x - 1 )\) is a factor of \(\mathrm { f } ( x )\).
2. Hence solve the equation \(4 \sin ^ { 3 } \theta + 4 \sin ^ { 2 } \theta + 7 \sin \theta - 5 = 0\) for \(0 ^ { \circ } \leq \theta \leq 360 ^ { \circ }\).

1 The polynomial \(\mathrm { p } ( x )\) is given by $$\mathrm { p } ( x ) = x ^ { 3 } - 4 x ^ { 2 } - 7 x + k$$ where \(k\) is a constant.

1. 1. Given that \(x + 2\) is a factor of \(\mathrm { p } ( x )\), show that \(k = 10\).
   2. Express \(\mathrm { p } ( x )\) as the product of three linear factors.
2. Use the Remainder Theorem to find the remainder when \(\mathrm { p } ( x )\) is divided by \(x - 3\).
3. Sketch the curve with equation \(y = x ^ { 3 } - 4 x ^ { 2 } - 7 x + 10\), indicating the values where the curve crosses the \(x\)-axis and the \(y\)-axis. (You are not required to find the coordinates of the stationary points.)

5

1. Factorise \(9 - 8 x - x ^ { 2 }\).
2. Show that \(25 - ( x + 4 ) ^ { 2 }\) can be written as \(9 - 8 x - x ^ { 2 }\).
3. A curve has equation \(y = 9 - 8 x - x ^ { 2 }\).
   1. Write down the equation of its line of symmetry.
   2. Find the coordinates of its vertex.
   3. Sketch the curve, indicating the values of the intercepts on the \(x\)-axis and the \(y\)-axis.

6

1. The polynomial \(\mathrm { p } ( x )\) is given by \(\mathrm { p } ( x ) = x ^ { 3 } - 7 x - 6\).
   1. Use the Factor Theorem to show that \(x + 1\) is a factor of \(\mathrm { p } ( x )\).
   2. Express \(\mathrm { p } ( x ) = x ^ { 3 } - 7 x - 6\) as the product of three linear factors.
2. The curve with equation \(y = x ^ { 3 } - 7 x - 6\) is sketched below. \includegraphics[max width=\textwidth, alt={}, center]{de4f827d-f237-488a-9177-3d85d0cb1771-4_403_762_651_641} The curve cuts the \(x\)-axis at the point \(A\) and the points \(B ( - 1,0 )\) and \(C ( 3,0 )\).
   1. State the coordinates of the point \(A\).
   2. Find \(\int _ { - 1 } ^ { 3 } \left( x ^ { 3 } - 7 x - 6 \right) \mathrm { d } x\).
   3. Hence find the area of the shaded region bounded by the curve \(y = x ^ { 3 } - 7 x - 6\) and the \(x\)-axis between \(B\) and \(C\).
   4. Find the gradient of the curve \(y = x ^ { 3 } - 7 x - 6\) at the point \(B\).
   5. Hence find an equation of the normal to the curve at the point \(B\).

6

1. The polynomial \(\mathrm { f } ( x )\) is given by \(\mathrm { f } ( x ) = x ^ { 3 } + 4 x - 5\).
   1. Use the Factor Theorem to show that \(x - 1\) is a factor of \(\mathrm { f } ( x )\).
   2. Express \(\mathrm { f } ( x )\) in the form \(( x - 1 ) \left( x ^ { 2 } + p x + q \right)\), where \(p\) and \(q\) are integers.
   3. Hence show that the equation \(\mathrm { f } ( x ) = 0\) has exactly one real root and state its value.
2. The curve with equation \(y = x ^ { 3 } + 4 x - 5\) is sketched below. \includegraphics[max width=\textwidth, alt={}, center]{23f34515-3373-4644-a8a1-82b45809d934-4_505_959_868_529} The curve cuts the \(x\)-axis at the point \(A ( 1,0 )\) and the point \(B ( 2,11 )\) lies on the curve.
   1. Find \(\int \left( x ^ { 3 } + 4 x - 5 \right) \mathrm { d } x\).
   2. Hence find the area of the shaded region bounded by the curve and the line \(A B\).

7 The quadratic equation $$( 2 k - 3 ) x ^ { 2 } + 2 x + ( k - 1 ) = 0$$ where \(k\) is a constant, has real roots.

1. Show that \(2 k ^ { 2 } - 5 k + 2 \leqslant 0\).
2. 1. Factorise \(2 k ^ { 2 } - 5 k + 2\).
   2. Hence, or otherwise, solve the quadratic inequality $$2 k ^ { 2 } - 5 k + 2 \leqslant 0$$

6 The polynomial \(\mathrm { p } ( x )\) is given by \(\mathrm { p } ( x ) = x ^ { 3 } + x ^ { 2 } - 8 x - 12\).

1. Use the Remainder Theorem to find the remainder when \(\mathrm { p } ( x )\) is divided by \(x - 1\).
2. 1. Use the Factor Theorem to show that \(x + 2\) is a factor of \(\mathrm { p } ( x )\).
   2. Express \(\mathrm { p } ( x )\) as the product of linear factors.
3. 1. The curve with equation \(y = x ^ { 3 } + x ^ { 2 } - 8 x - 12\) passes through the point \(( 0 , k )\). State the value of \(k\).
   2. Sketch the graph of \(y = x ^ { 3 } + x ^ { 2 } - 8 x - 12\), indicating the values of \(x\) where the curve touches or crosses the \(x\)-axis.

4

1. The polynomial \(\mathrm { p } ( x )\) is given by \(\mathrm { p } ( x ) = x ^ { 3 } - x + 6\).
   1. Find the remainder when \(\mathrm { p } ( x )\) is divided by \(x - 3\).
   2. Use the Factor Theorem to show that \(x + 2\) is a factor of \(\mathrm { p } ( x )\).
   3. Express \(\mathrm { p } ( x ) = x ^ { 3 } - x + 6\) in the form \(( x + 2 ) \left( x ^ { 2 } + b x + c \right)\), where \(b\) and \(c\) are integers.
   4. The equation \(\mathrm { p } ( x ) = 0\) has one root equal to - 2 . Show that the equation has no other real roots.
2. The curve with equation \(y = x ^ { 3 } - x + 6\) is sketched below. \includegraphics[max width=\textwidth, alt={}, center]{5f1ff5fa-b6e8-4c4f-aef7-63eb947b299f-3_529_702_945_667} The curve cuts the \(x\)-axis at the point \(A ( - 2,0 )\) and the \(y\)-axis at the point \(B\).
   1. State the \(y\)-coordinate of the point \(B\).
   2. Find \(\int _ { - 2 } ^ { 0 } \left( x ^ { 3 } - x + 6 \right) \mathrm { d } x\).
   3. Hence find the area of the shaded region bounded by the curve \(y = x ^ { 3 } - x + 6\) and the line \(A B\).

4. \includegraphics[max width=\textwidth, alt={}, center]{30d4e6e5-8235-44b0-ad8e-c4c0b313677f-1_572_803_1336_461} The diagram shows the curves with equations \(y = 7 - 2 x - 3 x ^ { 2 }\) and \(y = \frac { 2 } { x }\).
The two curves intersect at the points \(P , Q\) and \(R\).

1. Show that the \(x\)-coordinates of \(P , Q\) and \(R\) satisfy the equation $$3 x ^ { 3 } + 2 x ^ { 2 } - 7 x + 2 = 0$$ Given that \(P\) has coordinates \(( - 2 , - 1 )\),
2. find the coordinates of \(Q\) and \(R\).

1

1. The polynomial \(\mathrm { f } ( x )\) is defined by \(\mathrm { f } ( x ) = 3 x ^ { 3 } + 2 x ^ { 2 } - 7 x + 2\).
   1. Find f(1).
   2. Show that \(\mathrm { f } ( - 2 ) = 0\).
   3. Hence, or otherwise, show that $$\frac { ( x - 1 ) ( x + 2 ) } { 3 x ^ { 3 } + 2 x ^ { 2 } - 7 x + 2 } = \frac { 1 } { a x + b }$$ where \(a\) and \(b\) are integers.
2. The polynomial \(\mathrm { g } ( x )\) is defined by \(\mathrm { g } ( x ) = 3 x ^ { 3 } + 2 x ^ { 2 } - 7 x + d\). When \(\mathrm { g } ( x )\) is divided by \(( 3 x - 1 )\), the remainder is 2 . Find the value of \(d\).

2 The polynomial \(\mathrm { f } ( x )\) is defined by \(\mathrm { f } ( x ) = 2 x ^ { 3 } - 7 x ^ { 2 } + 13\).

1. Use the Remainder Theorem to find the remainder when \(\mathrm { f } ( x )\) is divided by \(( 2 x - 3 )\).
2. The polynomial \(\mathrm { g } ( x )\) is defined by \(\mathrm { g } ( x ) = 2 x ^ { 3 } - 7 x ^ { 2 } + 13 + d\), where \(d\) is a constant. Given that ( \(2 x - 3\) ) is a factor of \(\mathrm { g } ( x )\), show that \(d = - 4\).
3. Express \(\mathrm { g } ( x )\) in the form \(( 2 x - 3 ) \left( x ^ { 2 } + a x + b \right)\).

2

1. The polynomial \(\mathrm { f } ( x )\) is defined by \(\mathrm { f } ( x ) = 2 x ^ { 3 } + 3 x ^ { 2 } - 18 x + 8\).
   1. Use the Factor Theorem to show that \(( 2 x - 1 )\) is a factor of \(\mathrm { f } ( x )\).
   2. Write \(\mathrm { f } ( x )\) in the form \(( 2 x - 1 ) \left( x ^ { 2 } + p x + q \right)\), where \(p\) and \(q\) are integers.
   3. Simplify the algebraic fraction \(\frac { 4 x ^ { 2 } + 16 x } { 2 x ^ { 3 } + 3 x ^ { 2 } - 18 x + 8 }\).
2. Express the algebraic fraction \(\frac { 2 x ^ { 2 } } { ( x + 5 ) ( x - 3 ) }\) in the form \(A + \frac { B + C x } { ( x + 5 ) ( x - 3 ) }\), where \(A , B\) and \(C\) are integers.