Moderate -0.5 This is a straightforward application of the Remainder Theorem requiring students to substitute x=1 and x=-2 to create two simultaneous linear equations in a and b. The arithmetic is simple and the method is direct textbook procedure, making it easier than average but not trivial since it requires setting up and solving a system.
2.
$$f ( x ) = x ^ { 4 } - x ^ { 3 } + 3 x ^ { 2 } + a x + b$$
where \(a\) and \(b\) are constants.
When \(\mathrm { f } ( x )\) is divided by \(( x - 1 )\) the remainder is 4
When \(\mathrm { f } ( x )\) is divided by \(( x + 2 )\) the remainder is 22
Find the value of \(a\) and the value of \(b\).
Alternatively dividing \(f(x)\) by \((x-1)\) or \((x+2)\) and setting remainder equal to \(4\) or \(22\)
\(3 + a + b = 4\) or equivalent
A1
Powers must be multiplied out; no requirement to collect terms
\(16 + 8 + 12 - 2a + b = 22\) or equivalent
A1
Powers must be multiplied out; no requirement to collect terms
Solve simultaneous equations to obtain \(a = 5\) and \(b = -4\)
M1 A1
Correct simplified equations are \(a + b = 1\) and \(-2a + b = -14\). Minimal evidence required; accept values of both \(a\) and \(b\).
## Question 2:
| Answer | Mark | Guidance |
|--------|------|----------|
| Attempts $f(\pm 1) = 4$ **or** attempts $f(\pm 2) = 22$ | M1 | Alternatively dividing $f(x)$ by $(x-1)$ or $(x+2)$ and setting remainder equal to $4$ or $22$ |
| $3 + a + b = 4$ or equivalent | A1 | Powers must be multiplied out; no requirement to collect terms |
| $16 + 8 + 12 - 2a + b = 22$ or equivalent | A1 | Powers must be multiplied out; no requirement to collect terms |
| Solve simultaneous equations to obtain $a = 5$ and $b = -4$ | M1 A1 | Correct simplified equations are $a + b = 1$ and $-2a + b = -14$. Minimal evidence required; accept values of both $a$ and $b$. |
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2.
$$f ( x ) = x ^ { 4 } - x ^ { 3 } + 3 x ^ { 2 } + a x + b$$
where $a$ and $b$ are constants.\\
When $\mathrm { f } ( x )$ is divided by $( x - 1 )$ the remainder is 4\\
When $\mathrm { f } ( x )$ is divided by $( x + 2 )$ the remainder is 22\\
Find the value of $a$ and the value of $b$.\\
\hfill \mbox{\textit{Edexcel C12 2015 Q2 [5]}}