The polynomial P(x) is given by P(x) = x³ - 4x² - 7x + 10. (a) Show that (x-1) is a factor of P(x). (b) Hence, or otherwise, factorise P(x) completely. (c) Sketch the graph of y = P(x), showing the coordinates of the points where the graph crosses the coordinate axes.

Mark scheme

Marking points from the official mark scheme. The mark or mark type is shown in brackets.

1. (a) M1 for substituting x=1 into P(x)
2. (a) A1 for P(1) = (1)³ - 4(1)² - 7(1) + 10 = 1 - 4 - 7 + 10 = 0, leading to a conclusion that (x-1) is a factor.
3. (b) M1 for attempting polynomial division by (x-1) or by comparing coefficients
4. (b) A1 for correctly finding the quadratic factor (x² - 3x - 10)
5. (b) M1 for attempting to factorise the quadratic factor (x² - 3x - 10)
6. (b) A1 for P(x) = (x-1)(x-5)(x+2)
7. (c) B1 for a cubic shape with two turning points (correct general shape)
8. (c) B1 for showing correct x-intercepts at $(-2, 0), (1, 0), (5, 0)$
9. (c) B1 for showing correct y-intercept at $(0, 10)$
10. (c) B1 for graph starting from bottom left and ending at top right, consistent with positive coefficient of x³.