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

4

1. 1. Express \(16 - 6 x - x ^ { 2 }\) in the form \(p - ( x + q ) ^ { 2 }\) where \(p\) and \(q\) are integers.
   2. Hence write down the maximum value of \(16 - 6 x - x ^ { 2 }\).
2. 1. Factorise \(16 - 6 x - x ^ { 2 }\).
   2. Sketch the curve with equation \(y = 16 - 6 x - x ^ { 2 }\), stating the values of \(x\) where the curve crosses the \(x\)-axis and the value of the \(y\)-intercept.
      [0pt] [3 marks]

5 The polynomial \(\mathrm { p } ( x )\) is given by $$\mathrm { p } ( x ) = x ^ { 3 } + c x ^ { 2 } + d x + 3$$ where \(c\) and \(d\) are integers.

1. Given that \(x + 3\) is a factor of \(\mathrm { p } ( x )\), show that $$3 c - d = 8$$
2. The remainder when \(\mathrm { p } ( x )\) is divided by \(x - 2\) is 65 . Obtain a further equation in \(c\) and \(d\).
3. Use the equations from parts (a) and (b) to find the value of \(c\) and the value of \(d\). [3 marks]

6 The diagram shows a curve and a line which intersect at the points \(A , B\) and \(C\). \includegraphics[max width=\textwidth, alt={}, center]{f2124c89-79de-4758-b7b8-ff273345b9dd-7_574_844_349_609} The curve has equation \(y = x ^ { 3 } - x ^ { 2 } - 5 x + 7\) and the straight line has equation \(y = x + 7\). The point \(B\) has coordinates ( 0,7 ).

1. 1. Show that the \(x\)-coordinates of the points \(A\) and \(C\) satisfy the equation $$x ^ { 2 } - x - 6 = 0$$
   2. Find the coordinates of the points \(A\) and \(C\).
2. Find \(\int \left( x ^ { 3 } - x ^ { 2 } - 5 x + 7 \right) \mathrm { d } x\).
3. Find the area of the shaded region \(R\) bounded by the curve and the line segment \(A B\).
   [0pt] [4 marks] \(7 \quad\) A circle with centre \(C\) has equation \(x ^ { 2 } + y ^ { 2 } - 10 x + 12 y + 41 = 0\). The point \(A ( 3 , - 2 )\) lies on the circle.

7

1. Sketch the curve with equation \(y = x ^ { 2 } ( x - 3 )\).
2. The polynomial \(\mathrm { p } ( x )\) is given by \(\mathrm { p } ( x ) = x ^ { 2 } ( x - 3 ) + 20\).
   1. Find the remainder when \(\mathrm { p } ( x )\) is divided by \(x - 4\).
   2. Use the Factor Theorem to show that \(x + 2\) is a factor of \(\mathrm { p } ( x )\).
   3. Express \(\mathrm { p } ( x )\) in the form \(( x + 2 ) \left( x ^ { 2 } + b x + c \right)\), where \(b\) and \(c\) are integers.
   4. Hence show that the equation \(\mathrm { p } ( x ) = 0\) has exactly one real root and state its value.
      [0pt] [3 marks]

4 The polynomial \(\mathrm { p } ( x )\) is given by \(\mathrm { p } ( x ) = x ^ { 3 } - 5 x ^ { 2 } - 8 x + 48\).

1. 1. Use the Factor Theorem to show that \(x + 3\) is a factor of \(\mathrm { p } ( x )\).
   2. Express \(\mathrm { p } ( x )\) as a product of three linear factors.
2. 1. Use the Remainder Theorem to find the remainder when \(\mathrm { p } ( x )\) is divided by \(x - 2\).
   2. Express \(\mathrm { p } ( x )\) in the form \(( x - 2 ) \left( x ^ { 2 } + b x + c \right) + r\), where \(b , c\) and \(r\) are integers. [3 marks]

1. $$f ( x ) = ( \sqrt { x } + 3 ) ^ { 2 } + ( 1 - 3 \sqrt { x } ) ^ { 2 }$$ Show that \(\mathrm { f } ( x )\) can be written in the form \(a x + b\) where \(a\) and \(b\) are integers to be found.

5. $$f ( x ) = 4 x - 3 x ^ { 2 } - x ^ { 3 }$$

1. Fully factorise \(4 x - 3 x ^ { 2 } - x ^ { 3 }\).
2. Sketch the curve \(y = \mathrm { f } ( x )\), showing the coordinates of any points of intersection with the coordinate axes.

1. The \(n\)th term of a sequence is defined by

$$u _ { n } = n ^ { 2 } - 6 n + 11 , \quad n \geq 1 .$$ Given that the \(k\) th term of the sequence is 38 , find the value of \(k\).

9. A curve has the equation \(y = x ^ { 3 } - 5 x ^ { 2 } + 7 x\).

1. Show that the curve only crosses the \(x\)-axis at one point. The point \(P\) on the curve has coordinates \(( 3,3 )\).
2. Find an equation for the normal to the curve at \(P\), giving your answer in the form \(a x + b y = c\), where \(a , b\) and \(c\) are integers. The normal to the curve at \(P\) meets the coordinate axes at \(Q\) and \(R\).
3. Show that triangle \(O Q R\), where \(O\) is the origin, has area \(28 \frac { 1 } { 8 }\).

3

1. \(\quad\) Expand \(\left( x ^ { \frac { 3 } { 2 } } - 1 \right) ^ { 2 }\).
2. Hence find \(\int \left( x ^ { \frac { 3 } { 2 } } - 1 \right) ^ { 2 } \mathrm {~d} x\).
3. Hence find the value of \(\int _ { 1 } ^ { 4 } \left( x ^ { \frac { 3 } { 2 } } - 1 \right) ^ { 2 } \mathrm {~d} x\).

2. \(\mathrm { f } ( x ) = x ^ { 3 } + a x ^ { 2 } + b x - 10\), where \(a\) and \(b\) are constants. When \(\mathrm { f } ( x )\) is divided by \(( x - 3 )\), the remainder is 14 . When \(\mathrm { f } ( x )\) is divided by \(( x + 1 )\), the remainder is - 18 .

1. Find the value of \(a\) and the value of \(b\).
2. Show that \(( x - 2 )\) is a factor of \(\mathrm { f } ( x )\).
   [0pt] [P3 June 2002 Question 1]

7. \begin{figure}[h] \end{figure} Fig. 2 shows part of the curve \(C\) with equation \(y = \mathrm { f } ( x )\), where \(\mathrm { f } ( x ) = x ^ { 3 } - 6 x ^ { 2 } + 5 x\).
The curve crosses the \(x\)-axis at the origin \(O\) and at the points \(A\) and \(B\).

1. Factorise \(\mathrm { f } ( x )\) completely.
2. Write down the \(x\)-coordinates of the points \(A\) and \(B\).
3. Find the gradient of \(C\) at \(A\). The region \(R\) is bounded by \(C\) and the line \(O A\), and the region \(S\) is bounded by \(C\) and the line \(A B\).
4. Use integration to find the area of the combined regions \(R\) and \(S\), shown shaded in Fig.2. \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Figure 3} \includegraphics[alt={},max width=\textwidth]{e4d38009-aa70-4c55-9765-c54044ffaa31-4_736_727_338_402}
   \end{figure} Fig. 3 shows a sketch of part of the curve \(C\) with equation \(y = x ^ { 3 } - 7 x ^ { 2 } + 15 x + 3 , x \geq 0\). The point \(P\), on \(C\), has \(x\)-coordinate 1 and the point \(Q\) is the minimum turning point of \(C\).
   1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
   2. Find the coordinates of \(Q\).
   3. Show that \(P Q\) is parallel to the \(x\)-axis.
   4. Calculate the area, shown shaded in Fig. 3, bounded by \(C\) and the line \(P Q\).

1. \(\quad \mathrm { f } ( x ) = 3 x ^ { 3 } - 2 x ^ { 2 } + k x + 9\).

Given that when \(\mathrm { f } ( x )\) is divided by ( \(x + 2\) ) there is a remainder of - 35 ,

1. find the value of the constant \(k\),
2. find the remainder when \(\mathrm { f } ( x )\) is divided by ( \(3 x - 2\) ).
   

9. $$f ( x ) = x ^ { 3 } - 9 x ^ { 2 } + 24 x - 16 .$$

1. Evaluate \(\mathrm { f } ( 1 )\) and hence state a linear factor of \(\mathrm { f } ( x )\).
2. Show that \(\mathrm { f } ( x )\) can be expressed in the form $$\mathrm { f } ( x ) = ( x + p ) ( x + q ) ^ { 2 } ,$$ where \(p\) and \(q\) are integers to be found.
3. Sketch the curve \(y = \mathrm { f } ( x )\).
4. Using integration, find the area of the region enclosed by the curve \(y = \mathrm { f } ( x )\) and the \(x\)-axis.

9. The polynomial \(\mathrm { f } ( x )\) is given by $$f ( x ) = x ^ { 3 } + k x ^ { 2 } - 7 x - 15$$ where \(k\) is a constant.
When \(\mathrm { f } ( x )\) is divided by ( \(x + 1\) ) the remainder is \(r\).
When \(\mathrm { f } ( x )\) is divided by \(( x - 3 )\) the remainder is \(3 r\).

1. Find the value of \(k\).
2. Find the value of \(r\).
3. Show that \(( x - 5 )\) is a factor of \(\mathrm { f } ( x )\).
4. Show that there is only one real solution to the equation \(\mathrm { f } ( x ) = 0\). END

9. The first three terms of a geometric series are \(( x - 2 ) , ( x + 6 )\) and \(x ^ { 2 }\) respectively.

1. Show that \(x\) must be a solution of the equation $$x ^ { 3 } - 3 x ^ { 2 } - 12 x - 36 = 0$$
2. Verify that \(x = 6\) is a solution of equation (I) and show that there are no other real solutions. Using \(x = 6\),
3. find the common ratio of the series,
4. find the sum of the first eight terms of the series.

2. $$f ( x ) = x ^ { 3 } + k x - 20 .$$ Given that \(\mathrm { f } ( x )\) is exactly divisible by ( \(x + 1\) ),

1. find the value of the constant \(k\),
2. solve the equation \(\mathrm { f } ( x ) = 0\).

5. Given that $$f ( x ) = x ^ { 3 } + 7 x ^ { 2 } + p x - 6 ,$$ and that \(x = - 3\) is a solution to the equation \(\mathrm { f } ( x ) = 0\),

1. find the value of the constant \(p\),
2. show that when \(\mathrm { f } ( x )\) is divided by \(( x - 2 )\) there is a remainder of 50 ,
3. find the other solutions to the equation \(\mathrm { f } ( x ) = 0\), giving your answers to 2 decimal places.

9. \(f ( x ) = x ^ { 3 } - 4 x ^ { 2 } - 3 x + 18\).

1. Show that \(( x - 3 )\) is a factor of \(\mathrm { f } ( x )\).
2. Fully factorise \(\mathrm { f } ( x )\).
3. Using your answer to part (b), write down the coordinates of one of the turning points of the curve \(y = \mathrm { f } ( x )\) and give a reason for your answer.
4. Using differentiation, find the \(x\)-coordinate of the other turning point of the curve \(y = \mathrm { f } ( x )\).

6. The polynomial \(\mathrm { p } ( x )\) is defined by $$\mathrm { p } ( x ) = 2 x ^ { 3 } + x ^ { 2 } + a x + b ,$$ where \(a\) and \(b\) are constants.
Given that when \(\mathrm { p } ( x )\) is divided by \(( x + 2 )\) there is a remainder of 20 ,

1. find an expression for \(b\) in terms of \(a\). Given also that \(( x + 3 )\) is a factor of \(\mathrm { p } ( x )\),
2. find the values of \(a\) and \(b\),
3. fully factorise \(\mathrm { p } ( x )\).

5.

1. Show that \(( 2 x + 3 )\) is a factor of \(\left( 2 x ^ { 3 } - x ^ { 2 } + 4 x + 15 \right)\).
2. Hence, simplify $$\frac { 2 x ^ { 2 } + x - 3 } { 2 x ^ { 3 } - x ^ { 2 } + 4 x + 15 } .$$
3. Find the coordinates of the stationary points of the curve with equation $$y = \frac { 2 x ^ { 2 } + x - 3 } { 2 x ^ { 3 } - x ^ { 2 } + 4 x + 15 } .$$

2

1. The polynomial \(\mathrm { f } ( x )\) is defined by \(\mathrm { f } ( x ) = 9 x ^ { 3 } + 18 x ^ { 2 } - x - 2\).
   1. Use the Factor Theorem to show that \(3 x + 1\) is a factor of \(\mathrm { f } ( x )\).
   2. Express \(\mathrm { f } ( x )\) as a product of three linear factors.
   3. Simplify \(\frac { 9 x ^ { 3 } + 21 x ^ { 2 } + 6 x } { \mathrm { f } ( x ) }\).
2. When the polynomial \(9 x ^ { 3 } + p x ^ { 2 } - x - 2\) is divided by \(3 x - 2\), the remainder is - 4 . Find the value of the constant \(p\).

6

1. Use the Factor Theorem to show that \(4 x - 3\) is a factor of $$16 x ^ { 3 } + 11 x - 15$$
2. Given that \(x = \cos \theta\), show that the equation $$27 \cos \theta \cos 2 \theta + 19 \sin \theta \sin 2 \theta - 15 = 0$$ can be written in the form $$16 x ^ { 3 } + 11 x - 15 = 0$$
3. Hence show that the only solutions of the equation $$27 \cos \theta \cos 2 \theta + 19 \sin \theta \sin 2 \theta - 15 = 0$$ are given by \(\cos \theta = \frac { 3 } { 4 }\).

1 The polynomial \(\mathrm { f } ( x )\) is defined by \(\mathrm { f } ( x ) = 2 x ^ { 3 } + x ^ { 2 } - 8 x - 7\).

1. Use the Remainder Theorem to find the remainder when \(\mathrm { f } ( x )\) is divided by \(( 2 x + 1 )\).
   (2 marks)
2. The polynomial \(\mathrm { g } ( x )\) is defined by \(\mathrm { g } ( x ) = \mathrm { f } ( x ) + d\), where \(d\) is a constant.
   1. Given that \(( 2 x + 1 )\) is a factor of \(\mathrm { g } ( x )\), show that \(\mathrm { g } ( x ) = 2 x ^ { 3 } + x ^ { 2 } - 8 x - 4\).
      (1 mark)
   2. Given that \(\mathrm { g } ( x )\) can be written as \(\mathrm { g } ( x ) = ( 2 x + 1 ) \left( x ^ { 2 } + a \right)\), where \(a\) is an integer, express \(\mathrm { g } ( x )\) as a product of three linear factors.
   3. Hence, or otherwise, show that \(\frac { \mathrm { g } ( x ) } { 2 x ^ { 3 } - 3 x ^ { 2 } - 2 x } = p + \frac { q } { x }\), where \(p\) and \(q\) are integers.
      (3 marks)