Moderate -0.8 This is a straightforward application of the factor theorem to find roots of a cubic. Students need to test small integer values to find one factor (x=1 works), perform polynomial division, then solve the resulting quadratic. While it requires multiple steps, each is routine and the question explicitly guides students to find where the curve crosses the x-axis (i.e., solve the equation). This is easier than average as it's a standard textbook exercise with integer roots and no conceptual challenges.
4 In this question you must show detailed reasoning.
Find the coordinates of the points where the curve \(y = x ^ { 3 } - 2 x ^ { 2 } - 5 x + 6\) crosses the \(x\)-axis.
For \(x=1\): \(1^3-2(1)^2-5(1)+6=0\), so \(x=1\) is a root or \((x-1)\) is a factor
M1
AO 1.1a — Finding one root/factor by factor theorem or division to remainder of 0. Conclusion needed for M1
\((x-1)(x^2-x-6)\)
M1
AO 1.1 — Factorising to find quadratic factor or division (method seen) or factor theorem again to get different root/factor
\((x-1)(x-3)(x+2)\)
DM1
AO 1.1 — Completion, all 3 factors/roots seen, dep on previous M1. Might not be in same place
\((1,0),\ (3,0),\ (-2,0)\)
A1
AO 1.1 — All 3 points as coordinates or pairs of values. Dep on M3
Total: [4]
## Question 4:
| Answer | Marks | Guidance |
|--------|-------|----------|
| $x^3-2x^2-5x+6=0$ | | |
| For $x=1$: $1^3-2(1)^2-5(1)+6=0$, so $x=1$ is a root or $(x-1)$ is a factor | M1 | AO 1.1a — Finding one root/factor by factor theorem or division to remainder of 0. Conclusion needed for M1 |
| $(x-1)(x^2-x-6)$ | M1 | AO 1.1 — Factorising to find quadratic factor or division (method seen) or factor theorem again to get different root/factor |
| $(x-1)(x-3)(x+2)$ | DM1 | AO 1.1 — Completion, all 3 factors/roots seen, dep on previous M1. Might not be in same place |
| $(1,0),\ (3,0),\ (-2,0)$ | A1 | AO 1.1 — All 3 points as coordinates or pairs of values. Dep on M3 |
**Total: [4]**
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4 In this question you must show detailed reasoning.\\
Find the coordinates of the points where the curve $y = x ^ { 3 } - 2 x ^ { 2 } - 5 x + 6$ crosses the $x$-axis.
\hfill \mbox{\textit{OCR MEI Paper 3 2023 Q4 [4]}}