## Question 2(i):

| Answer | Mark | Guidance |
|--------|------|----------|
| $f(-3)$ used | M1 | |
| $-54 - 27 + 69 + 12 [= 0]$ isw | A1 | or M1 for correct division by $(x+3)$ or for the quadratic factor found by inspection and A1 for concluding that $x = -3$ is a root (may be earned later); A0 for concluding that $x = -3$ is a factor |
| Attempt at division by $(x+3)$ as far as $2x^3 + 6x^2$ in working | M1 | or inspection with at least two terms of three-term quadratic factor correct; or at least one further root found using remainder theorem |
| Correctly obtaining $2x^2 - 9x + 4$ | A1 | or stating further factor, found from using remainder theorem again |
| Factorising the correct quadratic factor | M1 | for factors giving two terms of quadratic correct or for factors fitting one error in quadratic formula or completing square; M0 for formula etc without factors found; allow for $(x-4)$ and $(x - \frac{1}{2})$ given as factors |
| $(2x-1)(x-4)[(x+3)]$ isw | A1 | allow $2(x - \frac{1}{2})$ instead of $(2x-1)$; condone inclusion of $= 0$; isw $(x - \frac{1}{2})$ as factor and/or roots found, even if stated as factors |

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## Question 2(ii):

| Answer | Mark | Guidance |
|--------|------|----------|
| Sketch of cubic right way up, with two turning points | B1 | 0 if stops at $x$-axis; ignore graph of $y = 4x + 12$; must not be ruled; no curving back (except condone between $x=0$ and $x=0.5$); condone some 'flicking out' at ends but not approaching more turning points; must continue beyond axes; allow max on $y$-axis or in 1st or 2nd quadrants; condone some doubling/feathering |
| Values of intersections on $x$-axis shown, correct $(-3, 0.5$ and $4)$ or ft from their factors or roots in (i) | B1 | on graph or nearby in this part; mark intent for intersections with both axes; allow if no graph; condone 3 on neg $x$-axis as slip for $-3$; condone e.g. 0.5 roughly halfway between their 0 and 1 marked on $x$-axis |
| 12 marked on $y$-axis | B1 | or $x=0, y=12$ seen in this part if consistent with graph drawn; allow if no graph, but e.g. B0 for graph with intersection on $-ve$ $y$-axis or nowhere near their indicated 12 |

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## Question 2(iii):

| Answer | Mark | Guidance |
|--------|------|----------|
| $2x^3 - 3x^2 - 23x + 12 = 4x + 12$ oe | M1 | or ft their factorised $f(x)$ |
| $2x^3 - 3x^2 - 27x [= 0]$ | A1 | after equating, allow A1 for cancelling $(x+3)$ factor on both sides and obtaining $2x^2 - 9x [=0]$; condone slip of $= y$ instead of $= 0$ |
| $[x](2x-9)(x+3) [= 0]$ | M1 | for linear factors of correct cubic, giving two terms correct; or for quadratic formula or completing square used on correct quadratic $2x^2 - 3x - 27 = 0$, condoning one error in formula etc; or after cancelling $(x+3)$ factor allow M1 for $x(2x-9)$ oe or obtaining $x=0$ or $9/2$ oe; M0 for e.g. quadratic formula used on cubic, unless recovery and all 3 roots given |
| $[x =]\ 0, -3$ and $9/2$ oe | A1 | need not be all stated together; e.g. $x=0$ may be earlier |

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