Moderate -0.8 This is a straightforward application of the factor and remainder theorems requiring two substitutions to form simultaneous equations. The algebra is routine with no conceptual challenges—easier than average A-level content.
2 The polynomial \(\mathrm { p } ( x )\) is defined by
$$\mathrm { p } ( x ) = x ^ { 3 } + a x ^ { 2 } + b x + 16$$
where \(a\) and \(b\) are constants. It is given that \(( x + 2 )\) is a factor of \(\mathrm { p } ( x )\) and that the remainder is 72 when \(\mathrm { p } ( x )\) is divided by \(( x - 2 )\).
Find the values of \(a\) and \(b\).
2 The polynomial $\mathrm { p } ( x )$ is defined by
$$\mathrm { p } ( x ) = x ^ { 3 } + a x ^ { 2 } + b x + 16$$
where $a$ and $b$ are constants. It is given that $( x + 2 )$ is a factor of $\mathrm { p } ( x )$ and that the remainder is 72 when $\mathrm { p } ( x )$ is divided by $( x - 2 )$.
Find the values of $a$ and $b$.\\
\hfill \mbox{\textit{CAIE P2 2020 Q2 [5]}}