OCR C2 2007 June — Question 9

Exam BoardOCR
ModuleC2 (Core Mathematics 2)
Year2007
SessionJune
TopicFactor & Remainder Theorem
TypeShow equation reduces to polynomial

9 The polynomial \(f ( x )\) is given by $$f ( x ) = x ^ { 3 } + 6 x ^ { 2 } + x - 4 .$$
  1. (a) Show that ( \(\mathrm { x } + 1\) ) is a factor of \(\mathrm { f } ( \mathrm { x } )\).
    (b) Hence find the exact roots of the equation \(f ( x ) = 0\).
  2. (a) Show that the equation $$2 \log _ { 2 } ( x + 3 ) + \log _ { 2 } x - \log _ { 2 } ( 4 x + 2 ) = 1$$ can be written in the form \(f ( x ) = 0\).
    (b) Explain why the equation $$2 \log _ { 2 } ( x + 3 ) + \log _ { 2 } x - \log _ { 2 } ( 4 x + 2 ) = 1$$ has only one real root and state the exact value of this root.