Moderate -0.8 This is a straightforward application of the factor and remainder theorems requiring students to set up two simultaneous equations (p(-2)=0 and p(-1)=2) and solve for a and b. It's routine algebraic manipulation with no conceptual challenges beyond direct theorem application, making it easier than average but not trivial due to the simultaneous equation solving.
The polynomial \(ax^3 + 5x^2 - 4x + b\), where \(a\) and \(b\) are constants, is denoted by p\((x)\). It is given that \((x + 2)\) is a factor of p\((x)\) and that when p\((x)\) is divided by \((x + 1)\) the remainder is 2.
Find the values of \(a\) and \(b\). [5]
Substitute x = – 2, equate result to zero and obtain a correct equation,
e.g. −8a+20+8+b=0
B1
Substitute x = – 1 and equate result to 2
M1
Obtain a correct equation, e.g. – a + 5 + 4 + b = 2
A1
Solve for a or for b
M1
Obtain a = 3 and b = – 4
A1
5
Answer
Marks
Guidance
Question
Answer
Marks
Question 2:
2 | Substitute x = – 2, equate result to zero and obtain a correct equation,
e.g. −8a+20+8+b=0 | B1
Substitute x = – 1 and equate result to 2 | M1
Obtain a correct equation, e.g. – a + 5 + 4 + b = 2 | A1
Solve for a or for b | M1
Obtain a = 3 and b = – 4 | A1
5
Question | Answer | Marks | Guidance
The polynomial $ax^3 + 5x^2 - 4x + b$, where $a$ and $b$ are constants, is denoted by p$(x)$. It is given that $(x + 2)$ is a factor of p$(x)$ and that when p$(x)$ is divided by $(x + 1)$ the remainder is 2.
Find the values of $a$ and $b$. [5]
\hfill \mbox{\textit{CAIE P3 2021 Q2 [5]}}