Moderate -0.3 This is a straightforward application of the factor and remainder theorems requiring two substitutions to create simultaneous equations. The arithmetic involves fractions but the method is standard textbook procedure with no conceptual challenges or novel problem-solving required.
3 The polynomial \(a x ^ { 3 } + x ^ { 2 } + b x + 3\) is denoted by \(\mathrm { p } ( x )\). It is given that \(\mathrm { p } ( x )\) is divisible by ( \(2 x - 1\) ) and that when \(\mathrm { p } ( x )\) is divided by \(( x + 2 )\) the remainder is 5 .
Find the values of \(a\) and \(b\).
3 The polynomial $a x ^ { 3 } + x ^ { 2 } + b x + 3$ is denoted by $\mathrm { p } ( x )$. It is given that $\mathrm { p } ( x )$ is divisible by ( $2 x - 1$ ) and that when $\mathrm { p } ( x )$ is divided by $( x + 2 )$ the remainder is 5 .
Find the values of $a$ and $b$.\\
\hfill \mbox{\textit{CAIE P3 2022 Q3 [5]}}