Standard +0.3 This is a straightforward application of the Factor and Remainder Theorem requiring students to set up two simultaneous equations from the given conditions and solve them. While it involves a quartic function, the algebra is routine with no conceptual challenges beyond standard C1 content, making it slightly easier than average.
The function \(f(x) = x^4 + bx + c\) is such that \(f(2) = 0\). Also, when \(f(x)\) is divided by \(x + 3\), the remainder is \(85\).
Find the values of \(b\) and \(c\). [5]
need not be simplified; condone 8 or 32 as first term if \(2^r\) not seen
\(81 - 3b + c = 85\) oe
B2
M1 for f(−3) seen or used, condoning one error except +3b – need not be simplified or for long division as far as obtaining \(x^3 - 3x^2\) in quotient
\(20 + 5b = 0\) oe
M1
for elimination of one variable, ft their equations in \(b\) and \(c\), condoning one error in rearrangement of their original equations or in one term in the elimination
\(b = -4\) and \(c = -8\)
A1
allow correct answers to imply last M1 after correct earlier equations
[5]
$16 + 2b + c = 0$ oe | M1 | need not be simplified; condone 8 or 32 as first term if $2^r$ not seen | in this question use annotation to indicate where part marks are earned
$81 - 3b + c = 85$ oe | B2 | M1 for f(−3) seen or used, condoning one error except +3b – need not be simplified or for long division as far as obtaining $x^3 - 3x^2$ in quotient | eg M1 for $81 - 3b + c = 0$ or 'long division' may be seen in grid or a mixture of methods may be used eg B2 for $c - (3b - 27) = 85$
$20 + 5b = 0$ oe | M1 | for elimination of one variable, ft their equations in $b$ and $c$, condoning one error in rearrangement of their original equations or in one term in the elimination | correct operation must be used in elimination
$b = -4$ and $c = -8$ | A1 | allow correct answers to imply last M1 after correct earlier equations | for misread of $x^4$ as $x^3$ or $x^2$ or higher powers, allow all 3 Ms equivalently
| [5] |
The function $f(x) = x^4 + bx + c$ is such that $f(2) = 0$. Also, when $f(x)$ is divided by $x + 3$, the remainder is $85$.
Find the values of $b$ and $c$. [5]
\hfill \mbox{\textit{OCR MEI C1 2012 Q8 [5]}}