Moderate -0.8 This is a straightforward polynomial division problem requiring students to either perform algebraic long division or expand and compare coefficients. It's a routine AS-level technique with clear methodology and no conceptual challenges—easier than average but not trivial since it requires careful algebraic manipulation.
6 Show that the expression \(3 x ^ { 3 } + x ^ { 2 } - 6 x - 5\) can be written in the form \(( x + 2 ) \left( a x ^ { 2 } + b x + c \right) + d\) where \(a\), \(b\), \(c\) and \(d\) are constants to be determined.
Allow grid method or long division as far as the linear term of the quotient.
\(b = -5\)
A1
May be embedded in final expression
\(c = 4\)
A1
May be embedded in final expression. FT their \(b\)
\(d = -13\)
B1
May also be found using the remainder theorem. May be embedded in final expression.
\(\text{f}(x) = (x+2)(3x^2 - 5x + 4) - 13\)
Need not be given explicitly if all coefficients seen
Alternative Method (preceding Q7):
Answer
Marks
Guidance
Answer
Marks
Guidance
\(a = 3\)
B1
Expanding and equating coefficients: quadratic term \(2a + b = 1\), linear term \(2b + c = -6\), constant term \(2c + d = -5\)
M1
Expanding and equating coefficients for at least the quadratic or linear term
\(b = -5\)
A1
May be embedded in final expression
\(c = 4\)
A1
May be embedded in final expression FT their \(b\)
\(d = -13\)
B1
May also be found using the remainder theorem. May be embedded in final expression
So \(f(x) = (x+2)(3x^2 - 5x + 4) - 13\)
Need not be given explicitly if all coefficients seen
## Question 6:
$a = 3$ | B1 |
Attempt to divide the cubic by $(x + 2)$ | M1 | Allow grid method or long division as far as the linear term of the quotient.
$b = -5$ | A1 | May be embedded in final expression
$c = 4$ | A1 | May be embedded in final expression. FT their $b$
$d = -13$ | B1 | May also be found using the remainder theorem. May be embedded in final expression.
$\text{f}(x) = (x+2)(3x^2 - 5x + 4) - 13$ | | Need not be given explicitly if all coefficients seen
## Alternative Method (preceding Q7):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $a = 3$ | B1 | |
| Expanding and equating coefficients: quadratic term $2a + b = 1$, linear term $2b + c = -6$, constant term $2c + d = -5$ | M1 | Expanding and equating coefficients for at least the quadratic or linear term |
| $b = -5$ | A1 | May be embedded in final expression |
| $c = 4$ | A1 | May be embedded in final expression FT their $b$ |
| $d = -13$ | B1 | May also be found using the remainder theorem. May be embedded in final expression |
| So $f(x) = (x+2)(3x^2 - 5x + 4) - 13$ | | Need not be given explicitly if all coefficients seen |
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6 Show that the expression $3 x ^ { 3 } + x ^ { 2 } - 6 x - 5$ can be written in the form $( x + 2 ) \left( a x ^ { 2 } + b x + c \right) + d$ where $a$, $b$, $c$ and $d$ are constants to be determined.
\hfill \mbox{\textit{OCR MEI AS Paper 1 2023 Q6 [5]}}