- \(\mathrm { f } ( x ) = x ^ { 3 } + a x ^ { 2 } - a x + 48\), where \(a\) is a constant
Given that \(\mathrm { f } ( - 6 ) = 0\)
- show that \(a = 4\)
- express \(\mathrm { f } ( x )\) as a product of two algebraic factors.
Given that \(2 \log _ { 2 } ( x + 2 ) + \log _ { 2 } x - \log _ { 2 } ( x - 6 ) = 3\)
- show that \(x ^ { 3 } + 4 x ^ { 2 } - 4 x + 48 = 0\)
- hence explain why
$$2 \log _ { 2 } ( x + 2 ) + \log _ { 2 } x - \log _ { 2 } ( x - 6 ) = 3$$
has no real roots.