Solve p(exponential) = 0

Questions where you solve equations like p(e^x) = 0, p(2^y) = 0, or p(3^t) = 0 by first solving p(x) = 0, then solving the resulting exponential equations using logarithms.

9 questions · Standard +0.2

1.02j Manipulate polynomials: expanding, factorising, division, factor theorem1.06g Equations with exponentials: solve a^x = b
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CAIE P2 2021 June Q7
9 marks Standard +0.3
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 \(\mathrm { p } \left( \mathrm { e } ^ { y } + \mathrm { e } ^ { - y } \right) = 0\).
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CAIE P2 2022 June Q5
7 marks Standard +0.3
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.
CAIE P2 2022 November Q4
7 marks Standard +0.3
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.
CAIE P3 2011 June Q4
7 marks Standard +0.3
4 The polynomial \(\mathrm { f } ( x )\) is defined by $$f ( x ) = 12 x ^ { 3 } + 25 x ^ { 2 } - 4 x - 12$$
  1. Show that \(\mathrm { f } ( - 2 ) = 0\) and factorise \(\mathrm { f } ( x )\) completely.
  2. Given that $$12 \times 27 ^ { y } + 25 \times 9 ^ { y } - 4 \times 3 ^ { y } - 12 = 0$$ state the value of \(3 ^ { y }\) and hence find \(y\) correct to 3 significant figures.
CAIE P2 2019 November Q4
7 marks Standard +0.3
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. Using this value of \(a\), factorise \(\mathrm { p } ( x )\) completely.
  3. Hence solve the equation \(\mathrm { p } \left( \mathrm { e } ^ { \sqrt { } y } \right) = 0\), giving the answer correct to 2 significant figures.
Edexcel C12 2017 January Q8
10 marks Standard +0.3
8. $$f ( x ) = 2 x ^ { 3 } - 5 x ^ { 2 } - 23 x - 10$$
  1. Find the remainder when \(\mathrm { f } ( x )\) is divided by ( \(x - 3\) ).
  2. Show that ( \(x + 2\) ) is a factor of \(\mathrm { f } ( x )\).
  3. Hence fully factorise \(\mathrm { f } ( x )\).
  4. Hence solve $$2 \left( 3 ^ { 3 t } \right) - 5 \left( 3 ^ { 2 t } \right) - 23 \left( 3 ^ { t } \right) = 10$$ giving your answer to 3 decimal places.
Edexcel C2 2013 June Q3
9 marks Standard +0.3
3. $$f ( x ) = 2 x ^ { 3 } - 5 x ^ { 2 } + a x + 18$$ where \(a\) is a constant. Given that \(( x - 3 )\) is a factor of \(\mathrm { f } ( x )\),
  1. show that \(a = - 9\)
  2. factorise \(\mathrm { f } ( x )\) completely. Given that $$\mathrm { g } ( y ) = 2 \left( 3 ^ { 3 y } \right) - 5 \left( 3 ^ { 2 y } \right) - 9 \left( 3 ^ { y } \right) + 18$$
  3. find the values of \(y\) that satisfy \(\mathrm { g } ( y ) = 0\), giving your answers to 2 decimal places where appropriate.
CAIE P2 2016 November Q4
8 marks Moderate -0.3
The polynomial \(\mathrm{p}(x)\) is defined by $$\mathrm{p}(x) = ax^3 + 3x^2 + 4ax - 5,$$ where \(a\) is a constant. It is given that \((2x - 1)\) is a factor of \(\mathrm{p}(x)\).
  1. Use the factor theorem to find the value of \(a\). [2]
  2. Factorise \(\mathrm{p}(x)\) and hence show that the equation \(\mathrm{p}(x) = 0\) has only one real root. [4]
  3. Use logarithms to solve the equation \(\mathrm{p}(6^x) = 0\) correct to 3 significant figures. [2]
AQA AS Paper 1 2020 June Q6
9 marks Moderate -0.3
  1. It is given that $$f(x) = x^3 - x^2 + x - 6$$ Use the factor theorem to show that \((x - 2)\) is a factor of \(f(x)\). [2 marks]
  2. Find the quadratic factor of \(f(x)\). [1 mark]
  3. Hence, show that there is only one real solution to \(f(x) = 0\) [3 marks]
  4. Find the exact value of \(x\) that solves $$e^{3x} - e^{2x} + e^x - 6 = 0$$ [3 marks]