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

670 questions

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Edexcel FP1 2010 June Q4
7 marks Moderate -0.8
4. $$f ( x ) = x ^ { 3 } + x ^ { 2 } + 44 x + 150$$ Given that \(\mathrm { f } ( x ) = ( x + 3 ) \left( x ^ { 2 } + a x + b \right)\), where \(a\) and \(b\) are real constants,
  1. find the value of \(a\) and the value of \(b\).
  2. Find the three roots of \(\mathrm { f } ( x ) = 0\).
  3. Find the sum of the three roots of \(\mathrm { f } ( x ) = 0\).
Edexcel FP1 2012 June Q1
5 marks Moderate -0.8
1. $$f ( x ) = 2 x ^ { 3 } - 6 x ^ { 2 } - 7 x - 4$$
  1. Show that \(\mathrm { f } ( 4 ) = 0\)
  2. Use algebra to solve \(\mathrm { f } ( x ) = 0\) completely.
Edexcel FP1 2013 June Q3
7 marks Moderate -0.3
3. Given that \(x = \frac { 1 } { 2 }\) is a root of the equation $$2 x ^ { 3 } - 9 x ^ { 2 } + k x - 13 = 0 , \quad k \in \mathbb { R }$$ find
  1. the value of \(k\),
  2. the other 2 roots of the equation.
Edexcel FP1 2015 June Q1
5 marks Moderate -0.5
1. $$f ( x ) = 9 x ^ { 3 } - 33 x ^ { 2 } - 55 x - 25$$ Given that \(x = 5\) is a solution of the equation \(\mathrm { f } ( x ) = 0\), use an algebraic method to solve \(\mathrm { f } ( x ) = 0\) completely.
(5)
Edexcel FP1 2018 June Q1
6 marks Moderate -0.5
1. $$f ( z ) = 2 z ^ { 3 } - 4 z ^ { 2 } + 15 z - 13$$ Given that \(\mathrm { f } ( z ) \equiv ( z - 1 ) \left( 2 z ^ { 2 } + a z + b \right)\), where \(a\) and \(b\) are real constants,
  1. find the value of \(a\) and the value of \(b\).
  2. Hence use algebra to find the three roots of the equation \(\mathrm { f } ( \mathrm { z } ) = 0\)
Edexcel FP1 Specimen Q8
9 marks Moderate -0.3
8. $$\mathrm { f } ( x ) \equiv 2 x ^ { 3 } - 5 x ^ { 2 } + p x - 5 , p \in \mathbb { R }$$ Given that \(1 - 2 \mathrm { i }\) is a complex solution of \(\mathrm { f } ( x ) = 0\),
  1. write down the other complex solution of \(\mathrm { f } ( x ) = 0\),
  2. solve the equation \(\mathrm { f } ( x ) = 0\),
  3. find the value of \(p\).
Edexcel C1 2008 June Q2
3 marks Easy -1.8
Factorise completely $$x ^ { 3 } - 9 x .$$
Edexcel C2 2005 June Q3
6 marks Moderate -0.8
  1. Use the factor theorem to show that \(( x + 4 )\) is a factor of \(2 x ^ { 3 } + x ^ { 2 } - 25 x + 12\).
  2. Factorise \(2 x ^ { 3 } + x ^ { 2 } - 25 x + 12\) completely.
Edexcel C2 2006 June Q4
8 marks Moderate -0.8
$$f ( x ) = 2 x ^ { 3 } + 3 x ^ { 2 } - 29 x - 60$$
  1. Find the remainder when \(\mathrm { f } ( x )\) is divided by \(( x + 2 )\).
  2. Use the factor theorem to show that \(( x + 3 )\) is a factor of \(\mathrm { f } ( x )\).
  3. Factorise \(\mathrm { f } ( x )\) completely.
Edexcel C2 2007 June Q2
6 marks Moderate -0.8
$$f ( x ) = 3 x ^ { 3 } - 5 x ^ { 2 } - 16 x + 12$$
  1. Find the remainder when \(\mathrm { f } ( x )\) is divided by ( \(x - 2\) ). Given that \(( x + 2 )\) is a factor of \(\mathrm { f } ( x )\),
  2. factorise \(\mathrm { f } ( x )\) completely.
Edexcel F1 2017 January Q2
7 marks Standard +0.3
The quadratic equation $$2 x ^ { 2 } - x + 3 = 0$$ has roots \(\alpha\) and \(\beta\).
Without solving the equation,
  1. write down the value of \(( \alpha + \beta )\) and the value of \(\alpha \beta\)
  2. find the value of \(\frac { 1 } { \alpha } + \frac { 1 } { \beta }\)
  3. find a quadratic equation which has roots $$\left( 2 \alpha - \frac { 1 } { \beta } \right) \text { and } \left( 2 \beta - \frac { 1 } { \alpha } \right)$$ giving your answer in the form \(p x ^ { 2 } + q x + r = 0\) where \(p , q\) and \(r\) are integers.
OCR C1 2006 January Q2
5 marks Easy -1.2
2
  1. Simplify \(( 3 x + 1 ) ^ { 2 } - 2 ( 2 x - 3 ) ^ { 2 }\).
  2. Find the coefficient of \(x ^ { 3 }\) in the expansion of $$\left( 2 x ^ { 3 } - 3 x ^ { 2 } + 4 x - 3 \right) \left( x ^ { 2 } - 2 x + 1 \right)$$
OCR C1 2005 June Q4
5 marks Standard +0.3
4 Solve the equation \(x ^ { 6 } + 26 x ^ { 3 } - 27 = 0\).
OCR C1 2005 June Q6
7 marks Easy -1.2
6 Given that \(\mathrm { f } ( x ) = ( x + 1 ) ^ { 2 } ( 3 x - 4 )\),
  1. express \(\mathrm { f } ( x )\) in the form \(a x ^ { 3 } + b x ^ { 2 } + c x + d\),
  2. find \(\mathrm { f } ^ { \prime } ( x )\),
  3. find \(\mathrm { f } ^ { \prime \prime } ( x )\).
OCR C1 2007 June Q1
3 marks Easy -1.8
1 Simplify \(( 2 x + 5 ) ^ { 2 } - ( x - 3 ) ^ { 2 }\), giving your answer in the form \(a x ^ { 2 } + b x + c\).
OCR C1 2008 June Q6
6 marks Moderate -0.8
6
  1. Expand and simplify \(( x - 5 ) ( x + 2 ) ( x + 5 )\).
  2. Sketch the curve \(y = ( x - 5 ) ( x + 2 ) ( x + 5 )\), giving the coordinates of the points where the curve crosses the axes.
OCR C1 Specimen Q8
14 marks Moderate -0.3
8
  1. Find the coordinates of the stationary points on the curve \(y = 2 x ^ { 3 } - 3 x ^ { 2 } - 12 x - 7\).
  2. Determine whether each stationary point is a maximum point or a minimum point.
  3. By expanding the right-hand side, show that $$2 x ^ { 3 } - 3 x ^ { 2 } - 12 x - 7 = ( x + 1 ) ^ { 2 } ( 2 x - 7 )$$
  4. Sketch the curve \(y = 2 x ^ { 3 } - 3 x ^ { 2 } - 12 x - 7\), marking the coordinates of the stationary points and the points where the curve meets the axes.
OCR MEI C1 2008 January Q6
3 marks Moderate -0.8
6 When \(x ^ { 3 } + k x + 7\) is divided by \(( x - 2 )\), the remainder is 3 . Find the value of \(k\).
OCR MEI C1 2009 January Q4
3 marks Moderate -0.8
4 You are given that \(\mathrm { f } ( x ) = x ^ { 4 } + a x - 6\) and that \(x - 2\) is a factor of \(\mathrm { f } ( x )\).
Find the value of \(a\).
OCR MEI C1 2009 January Q5
5 marks Easy -1.2
5
  1. Find the coefficient of \(x ^ { 3 }\) in the expansion of \(\left( x ^ { 2 } - 3 \right) \left( x ^ { 3 } + 7 x + 1 \right)\).
  2. Find the coefficient of \(x ^ { 2 }\) in the binomial expansion of \(( 1 + 2 x ) ^ { 7 }\).
OCR MEI C1 2007 June Q4
3 marks Moderate -0.8
4 You are given that \(\mathrm { f } ( x ) = x ^ { 3 } + k x + c\). The value of \(\mathrm { f } ( 0 )\) is 6, and \(x - 2\) is a factor of \(\mathrm { f } ( x )\).
Find the values of \(k\) and \(c\).
OCR MEI C1 2007 June Q10
5 marks Moderate -0.8
10 The triangle shown in Fig. 10 has height \(( x + 1 ) \mathrm { cm }\) and base \(( 2 x - 3 ) \mathrm { cm }\). Its area is \(9 \mathrm {~cm} ^ { 2 }\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{4d8caf0f-7594-42cb-bd40-e6c11e2b6832-3_444_1088_351_715} \captionsetup{labelformat=empty} \caption{Fig. 10}
\end{figure}
  1. Show that \(2 x ^ { 2 } - x - 21 = 0\).
  2. By factorising, solve the equation \(2 x ^ { 2 } - x - 21 = 0\). Hence find the height and base of the triangle. Section B (36 marks)
OCR MEI C1 2007 June Q13
12 marks Moderate -0.3
13 A cubic polynomial is given by \(\mathrm { f } ( x ) = 2 x ^ { 3 } - x ^ { 2 } - 11 x - 12\).
  1. Show that \(( x - 3 ) \left( 2 x ^ { 2 } + 5 x + 4 \right) = 2 x ^ { 3 } - x ^ { 2 } - 11 x - 12\). Hence show that \(\mathrm { f } ( x ) = 0\) has exactly one real root.
  2. Show that \(x = 2\) is a root of the equation \(\mathrm { f } ( x ) = - 22\) and find the other roots of this equation.
  3. Using the results from the previous parts, sketch the graph of \(y = \mathrm { f } ( x )\).
OCR MEI C1 2008 June Q11
12 marks Moderate -0.3
11 You are given that \(\mathrm { f } ( x ) = 2 x ^ { 3 } + 7 x ^ { 2 } - 7 x - 12\).
  1. Verify that \(x = - 4\) is a root of \(\mathrm { f } ( x ) = 0\).
  2. Hence express \(\mathrm { f } ( x )\) in fully factorised form.
  3. Sketch the graph of \(y = \mathrm { f } ( x )\).
  4. Show that \(\mathrm { f } ( x - 4 ) = 2 x ^ { 3 } - 17 x ^ { 2 } + 33 x\).
OCR MEI C1 2015 June Q9
4 marks Moderate -0.8
9 Explain why each of the following statements is false. State in each case which of the symbols ⇒, ⟸ or ⇔ would make the statement true.
  1. ABCD is a square \(\Leftrightarrow\) the diagonals of quadrilateral ABCD intersect at \(90 ^ { \circ }\)
  2. \(x ^ { 2 }\) is an integer \(\Rightarrow x\) is an integer