Moderate -0.8 This is a straightforward application of the factor and remainder theorems requiring students to set up two simultaneous equations (f(-1)=0 and f(-2)=-5) and solve for a and b. It's a standard textbook exercise with clear conditions and routine algebraic manipulation, making it easier than average but not trivial since it requires understanding the theorems and solving simultaneous equations.
2 The cubic polynomial \(2 x ^ { 3 } + a x ^ { 2 } + b\) is denoted by \(\mathrm { f } ( x )\). It is given that ( \(x + 1\) ) is a factor of \(\mathrm { f } ( x )\), and that when \(\mathrm { f } ( x )\) is divided by \(( x + 2 )\) the remainder is - 5 . Find the values of \(a\) and \(b\).
Solve a relevant pair of equations, obtaining \(a\) or \(b\)
M1
Obtain both answers \(a = 3\) and \(b = -1\)
A1
5
State or obtain $-2 + a + b = 0$, or equivalent | B1 |
Substitute $x = -2$ and equate to $-5$ | M1 |
Obtain 3-term equation, or equivalent | A1 |
Solve a relevant pair of equations, obtaining $a$ or $b$ | M1 |
Obtain both answers $a = 3$ and $b = -1$ | A1 | 5 |
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2 The cubic polynomial $2 x ^ { 3 } + a x ^ { 2 } + b$ is denoted by $\mathrm { f } ( x )$. It is given that ( $x + 1$ ) is a factor of $\mathrm { f } ( x )$, and that when $\mathrm { f } ( x )$ is divided by $( x + 2 )$ the remainder is - 5 . Find the values of $a$ and $b$.
\hfill \mbox{\textit{CAIE P2 2002 Q2 [5]}}