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 for A-level.
4 The polynomial \(2 x ^ { 3 } + 7 x ^ { 2 } + a x + b\), where \(a\) and \(b\) are constants, is denoted by \(\mathrm { p } ( x )\). It is given that \(( x + 1 )\) is a factor of \(\mathrm { p } ( x )\), and that when \(\mathrm { p } ( x )\) is divided by \(( x + 2 )\) the remainder is 5 . Find the values of \(a\) and \(b\).
Substitute \(x = -1\), equate to zero and obtain a correct equation in any form
B1
Substitute \(x = -2\) and equate to 5
M1
Obtain a correct equation in any form
A1
Solve a relevant pair of equations for \(a\) or for \(b\)
M1
Obtain \(a = 2\) and \(b = -3\)
A1
[5]
Substitute $x = -1$, equate to zero and obtain a correct equation in any form | B1 |
Substitute $x = -2$ and equate to 5 | M1 |
Obtain a correct equation in any form | A1 |
Solve a relevant pair of equations for $a$ or for $b$ | M1 |
Obtain $a = 2$ and $b = -3$ | A1 |
| [5] |
4 The polynomial $2 x ^ { 3 } + 7 x ^ { 2 } + a x + b$, where $a$ and $b$ are constants, is denoted by $\mathrm { p } ( x )$. It is given that $( x + 1 )$ is a factor of $\mathrm { p } ( x )$, 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 P2 2008 Q4 [5]}}