Solving Equations via Substitution After Division

Questions that use polynomial division or factorisation results to solve equations involving exponential or trigonometric substitutions (e.g., solving for y where x = e^(3y) or x = cosec(2θ)).

2 questions · Standard +0.3

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CAIE P2 2021 June Q5
7 marks Standard +0.3
5
  1. Find the quotient when \(x ^ { 4 } - 32 x + 55\) is divided by \(( x - 2 ) ^ { 2 }\) and show that the remainder is 7 .
  2. Factorise \(x ^ { 4 } - 32 x + 48\).
  3. Hence solve the equation \(\mathrm { e } ^ { - 12 y } - 32 \mathrm { e } ^ { - 3 y } + 48 = 0\), giving your answer in an exact form.
CAIE P2 2022 March Q6
10 marks Standard +0.3
6 The polynomial \(\mathrm { p } ( x )\) is defined by $$\mathrm { p } ( x ) = 4 x ^ { 3 } + 16 x ^ { 2 } + 9 x - 15$$
  1. Find the quotient when \(\mathrm { p } ( x )\) is divided by \(( 2 x + 3 )\), and show that the remainder is - 6 .
  2. Find \(\int \frac { \mathrm { p } ( x ) } { 2 x + 3 } \mathrm {~d} x\).
  3. Factorise \(\mathrm { p } ( x ) + 6\) completely and hence solve the equation $$p ( \operatorname { cosec } 2 \theta ) + 6 = 0$$ for \(0 ^ { \circ } < \theta < 135 ^ { \circ }\).