Two unknowns with show-that step

Part (a) requires proving a relationship between the unknowns using one remainder condition, then part (b) uses the second remainder to solve the system.

12 questions · Moderate -0.4

1.02j Manipulate polynomials: expanding, factorising, division, factor theorem
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Edexcel C12 2016 June Q7
10 marks Moderate -0.3
7. $$f ( x ) = 3 x ^ { 3 } + a x ^ { 2 } + b x - 10 \text {, where } a \text { and } b \text { are constants. }$$ Given that \(( x - 2 )\) is a factor of \(\mathrm { f } ( x )\),
  1. use the factor theorem to show that \(2 a + b = - 7\) Given also that when \(\mathrm { f } ( x )\) is divided by \(( x + 1 )\) the remainder is - 36
  2. find the value of \(a\) and the value of \(b\). \(\mathrm { f } ( x )\) can be written in the form $$\mathrm { f } ( x ) = ( x - 2 ) \mathrm { Q } ( x ) \text {, where } \mathrm { Q } ( x ) \text { is a quadratic function. }$$
    1. Find \(\mathrm { Q } ( x )\).
    2. Prove that the equation \(\mathrm { f } ( x ) = 0\) has only one real root. You must justify your answer and show all your working.
Edexcel C12 2018 June Q2
6 marks Moderate -0.3
2. $$f ( x ) = a x ^ { 3 } + 2 x ^ { 2 } + b x - 3$$ where \(a\) and \(b\) are constants.
When \(\mathrm { f } ( x )\) is divided by ( \(2 x - 1\) ) the remainder is 1
  1. Show that $$a + 4 b = 28$$ When \(\mathrm { f } ( x )\) is divided by \(( x + 1 )\) the remainder is - 17
  2. Find the value of \(a\) and the value of \(b\).
Edexcel C12 2018 October Q8
9 marks Moderate -0.3
8. $$f ( x ) = 2 x ^ { 3 } - 3 x ^ { 2 } + p x + q$$ where \(p\) and \(q\) are constants.
When \(\mathrm { f } ( x )\) is divided by \(( x - 1 )\), the remainder is - 6
  1. Use the remainder theorem to show that \(p + q = - 5\) Given also that \(( x + 2 )\) is a factor of \(\mathrm { f } ( x )\),
  2. find the value of \(p\) and the value of \(q\).
  3. Factorise \(\mathrm { f } ( \mathrm { x } )\) completely.
Edexcel C12 Specimen Q6
7 marks Moderate -0.5
6. $$\mathrm { f } ( x ) = x ^ { 4 } + x ^ { 3 } + 2 x ^ { 2 } + a x + b ,$$ where \(a\) and \(b\) are constants. When \(\mathrm { f } ( x )\) is divided by \(( x - 1 )\), the remainder is 7
  1. Show that \(a + b = 3\) When \(\mathrm { f } ( x )\) is divided by ( \(x + 2\) ), the remainder is - 8
  2. Find the value of \(a\) and the value of \(b\).
Edexcel P2 2021 January Q1
6 marks Moderate -0.3
1. $$f ( x ) = x ^ { 4 } + a x ^ { 3 } - 3 x ^ { 2 } + b x + 5$$ where \(a\) and \(b\) are constants.
When \(\mathrm { f } ( x )\) is divided by ( \(x + 1\) ), the remainder is 4
  1. Show that \(a + b = - 1\) When \(\mathrm { f } ( x )\) is divided by ( \(x - 2\) ), the remainder is - 23
  2. Find the value of \(a\) and the value of \(b\).
Edexcel P2 2018 Specimen Q1
7 marks Moderate -0.8
1. $$\mathrm { f } ( x ) = x ^ { 4 } + x ^ { 3 } + 2 x ^ { 2 } + a x + b ,$$ where \(a\) and \(b\) are constants.
When \(\mathrm { f } ( x )\) is divided by ( \(x - 1\) ), the remainder is 7
  1. Show that \(a + b = 3\) When \(\mathrm { f } ( x )\) is divided by ( \(x + 2\) ), the remainder is - 8
  2. Find the value of \(a\) and the value of \(b\)
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Edexcel C2 2011 January Q1
7 marks Moderate -0.3
1. $$\mathrm { f } ( x ) = x ^ { 4 } + x ^ { 3 } + 2 x ^ { 2 } + a x + b$$ where \(a\) and \(b\) are constants. When \(\mathrm { f } ( x )\) is divided by \(( x - 1 )\), the remainder is 7 .
  1. Show that \(a + b = 3\). When \(\mathrm { f } ( x )\) is divided by \(( x + 2 )\), the remainder is - 8 .
  2. Find the value of \(a\) and the value of \(b\).
Edexcel C2 2012 January Q5
6 marks Moderate -0.3
  1. \(\mathrm { f } ( x ) = x ^ { 3 } + a x ^ { 2 } + b x + 3\), where \(a\) and \(b\) are constants.
Given that when \(\mathrm { f } ( x )\) is divided by \(( x + 2 )\) the remainder is 7 ,
  1. show that \(2 a - b = 6\) Given also that when \(\mathrm { f } ( x )\) is divided by \(( x - 1 )\) the remainder is 4 ,
  2. find the value of \(a\) and the value of \(b\).
Edexcel C2 2015 June Q3
9 marks Moderate -0.3
3. \(\mathrm { f } ( x ) = 6 x ^ { 3 } + 3 x ^ { 2 } + A x + B\), where \(A\) and \(B\) are constants. Given that when \(\mathrm { f } ( x )\) is divided by \(( x + 1 )\) the remainder is 45 ,
  1. show that \(B - A = 48\) Given also that ( \(2 x + 1\) ) is a factor of \(\mathrm { f } ( x )\),
  2. find the value of \(A\) and the value of \(B\).
  3. Factorise f(x) fully.
Edexcel C2 2018 June Q3
8 marks Moderate -0.3
3. $$f ( x ) = 24 x ^ { 3 } + A x ^ { 2 } - 3 x + B$$ where \(A\) and \(B\) are constants.
When \(\mathrm { f } ( x )\) is divided by \(( 2 x - 1 )\) the remainder is 30
  1. Show that \(A + 4 B = 114\) Given also that ( \(x + 1\) ) is a factor of \(\mathrm { f } ( x )\),
  2. find another equation in \(A\) and \(B\).
  3. Find the value of \(A\) and the value of \(B\).
  4. Hence find a quadratic factor of \(\mathrm { f } ( x )\).
OCR C2 Q8
10 marks Moderate -0.3
8. 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 \(( 2 x - 1 )\) is a factor of \(\mathrm { p } ( x )\),
  2. find the values of \(a\) and \(b\),
  3. fully factorise \(\mathrm { p } ( x )\).
Edexcel PURE 2024 October Q3
Moderate -0.8
3. $$f ( x ) = 2 x ^ { 3 } - x ^ { 2 } + A x + B$$ where \(A\) and \(B\) are integers.
Given that when \(\mathrm { f } ( x )\) is divided by \(( x + 3 )\) the remainder is 55
  1. show that $$3 A - B = - 118$$ Given also that \(( 2 x - 5 )\) is a factor of \(\mathrm { f } ( x )\),
  2. find the value of \(A\) and the value of \(B\).
  3. Hence find the quotient when \(\mathrm { f } ( x )\) is divided by ( \(x - 7\) )