Verify, factorise, solve with substitution

Questions that verify a given factor, factorise the cubic, then solve an equation where x is replaced by a trigonometric or other function (e.g., f(cosec θ) = 0).

4 questions · Standard +0.1

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

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CAIE P2 2020 June Q6

8 marks Standard +0.3

6 The polynomial \(\mathrm { p } ( x )\) is defined by $$\mathrm { p } ( x ) = 6 x ^ { 3 } + a x ^ { 2 } - 4 x - 3$$ where \(a\) is a constant. It is given that \(( x + 3 )\) is a factor of \(\mathrm { p } ( x )\).

Find the value of \(a\).
Using this value of \(a\), factorise \(\mathrm { p } ( x )\) completely.
Hence solve the equation \(\mathrm { p } ( \operatorname { cosec } \theta ) = 0\) for \(0 ^ { \circ } < \theta < 360 ^ { \circ }\).

CAIE P2 2024 November Q4

8 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 \(\mathrm { p } ( x )\).

Find the value of \(a\).
Hence factorise \(\mathrm { p } ( x )\) completely.
Solve the equation \(\mathrm { p } \left( \operatorname { cosec } ^ { 2 } \theta \right) = 0\) for \(- 90 ^ { \circ } < \theta < 90 ^ { \circ }\).

AQA Paper 1 2018 June Q12

10 marks Standard +0.3

Prove that ( \(2 x + 1\) ) is a factor of \(\mathrm { p } ( x )\) 12
Factorise \(\mathrm { p } ( x )\) completely.
12
Prove that there are no real solutions to the equation $$\frac { 30 \sec ^ { 2 } x + 2 \cos x } { 7 } = \sec x + 1$$

CAIE P2 2024 March Q3

7 marks Moderate -0.3

The polynomial \(\mathrm{p}(x)\) is defined by $$\mathrm{p}(x) = 6x^3 + ax^2 + 3x - 10,$$ where \(a\) is a constant. It is given that \((2x - 1)\) is a factor of \(\mathrm{p}(x)\).

Find the value of \(a\) and hence factorise \(\mathrm{p}(x)\) completely. [5]
Solve the equation \(\mathrm{p}(\cos\theta) = 0\) for \(-90° < \theta < 90°\). [2]