# Question 3:

## Part (i)
| Answer | Mark | Guidance |
|--------|------|----------|
| Sketch of cubic the right way up, with two turning points and clearly crossing $x$-axis in 3 places | B1 | No section to be ruled; no curving back; condone slight "flicking out" at ends; curve must clearly extend beyond $x$-axis at both ends |
| Crossing/reaching $x$-axis at $-4$, $-2$ and $1.5$ | B1 | Intersections must be shown correctly labelled or worked out nearby; accept curve crossing axis halfway between 1 and 2 if $\frac{3}{2}$ not marked |
| Intersection of $y$-axis at $-24$ | B1 | NB to find $-24$ some are expanding $f(x)$ here, which gains M1 in (iii)A |

**[3 marks]**

## Part (ii)
| Answer | Mark | Guidance |
|--------|------|----------|
| $-2$, $0$ and $\frac{7}{2}$ oe isw or ft their intersections | 2 | B1 for 2 correct or ft or for $(-2,0)$, $(0,0)$ and $(3.5, 0)$; or M1 for $(x+2)x(2x-7)$ oe; or SC1 for $-6$, $-4$ and $-\frac{1}{2}$ oe |

**[2 marks]**

## Part (iii)(A)
| Answer | Mark | Guidance |
|--------|------|----------|
| Correct expansion of product of 2 brackets of $f(x)$ | M1 | Need not be simplified; condone lack of brackets; or allow M1 for expansion of all 3 brackets showing all terms with at most one error: $2x^3 + 4x^2 + 8x^2 - 3x^2 + 16x - 12x - 6x - 24$; e.g. $2x^2+5x-12$ or $2x^2+x-6$ or $x^2+6x+8$ |
| Correct expansion of quadratic and linear, completion to given answer | A1 | For correct completion if all 3 brackets already expanded, with reference to show why $-24$ changes to $-9$; condone lack of brackets if correctly expanded; NB answer given, mark the whole process |

**[2 marks]**

## Question 3(iii)(B):

$g(1) = 2 + 9 - 2 - 9 = 0$ | B1 | Allow this mark for $(x-1)$ shown to be a factor and a statement that $x=1$ is a root [of $g(x)=0$]; B0 for just $g(1) = 2(1)^3 + 9(1)^2 - 2(1) - 9 [=0]$

Attempt at division by $(x-1)$ as far as $2x^3 - 2x^2$ in working | M1 | Or inspection with at least two terms of quadratic factor correct; M0 for division by $x+1$ after $g(1)=0$ unless further working such as $g(-1)=0$ shown, but this can go on to gain last M1A1

Correctly obtaining $2x^2 + 11x + 9$ | A1 | Allow B2 for another linear factor found by the factor theorem; NB mixture of methods may be seen

Factorising a correct quadratic factor | M1 | For factors giving two terms correct; e.g. allow M1 for factorising $2x^2 + 7x - 9$ after division by $x+1$; allow M1 for $(x+1)(x+\frac{18}{4})$ after $-1$ and $-\frac{18}{4}$ correctly found by formula

$(2x+9)(x+1)(x-1)$ isw | A1 | Allow $2(x+\frac{9}{2})(x+1)(x-1)$; dependent on 2nd M1 only; condone omission of first factor found; ignore $=0$ seen; SC alternative method: allow first M1A1 for $(2x+9)(x^2-1)$ then second M1A1 for full factorisation

**[5]**

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