Standard +0.3 This is a structured multi-part question that guides students through standard C2 techniques: applying factor theorem (routine), factorizing a cubic and solving (standard), and manipulating logarithms to reach a given polynomial (methodical but not demanding). Part (c) requires careful log manipulation but the target equation is given, and part (d) is trivial once (b) and (c) are complete. Slightly above average due to the multi-step nature and logarithm work, but all techniques are standard C2 fare with significant scaffolding.
7. (a) Use the factor theorem to show that \(( x + 1 )\) is a factor of \(x ^ { 3 } - x ^ { 2 } - 10 x - 8\).
(b) Find all the solutions of the equation \(x ^ { 3 } - x ^ { 2 } - 10 x - 8 = 0\).
(c) Prove that the value of \(x\) that satisfies
$$2 \log _ { 2 } x + \log _ { 2 } ( x - 1 ) = 1 + \log _ { 2 } ( 5 x + 4 )$$
is a solution of the equation
$$x ^ { 3 } - x ^ { 2 } - 10 x - 8 = 0$$
(d) State, with a reason, the value of \(x\) that satisfies equation (I).
7. (a) Use the factor theorem to show that $( x + 1 )$ is a factor of $x ^ { 3 } - x ^ { 2 } - 10 x - 8$.\\
(b) Find all the solutions of the equation $x ^ { 3 } - x ^ { 2 } - 10 x - 8 = 0$.\\
(c) Prove that the value of $x$ that satisfies
$$2 \log _ { 2 } x + \log _ { 2 } ( x - 1 ) = 1 + \log _ { 2 } ( 5 x + 4 )$$
is a solution of the equation
$$x ^ { 3 } - x ^ { 2 } - 10 x - 8 = 0$$
(d) State, with a reason, the value of $x$ that satisfies equation (I).\\
\hfill \mbox{\textit{Edexcel C2 Q7 [12]}}