A-LevelMathematicsNumerical methods9709/21 · Oct/Nov 2018 · Q48 Marks

The diagram shows the curve with equation y = x⁴ + 2x³ + 2x² – 12x – 32. The curve crosses the x-axis at points with coordinates (α, 0) and (β, 0). [Figure 6.1]

(i)Use the factor theorem to show that (x + 2) is a factor of x⁴ + 2x³ + 2x² – 12x – 32.[2]
(ii)Show that β satisfies an equation of the form x = ³√(p + qx), and state the values of p and q.[3]
(iii)Use an iterative formula based on the equation in part (ii) to find the value of β correct to 4 significant figures. Give the result of each iteration to 6 significant figures.[3]

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Mark scheme

Marking points from the official mark scheme (Oct/Nov 2018). The mark or mark type is shown in brackets.

[M1] Substitute -2 and simplify
[A1] Obtain 16 – 16 + 8 + 24 – 32 and hence zero and conclude AG; necessary detail needed
[M1] Attempt division by x+2 to reach at least partial quotient x³ + kx or use of identity or inspection
[A1] Obtain x³ + 2x – 16
[A1] Equate to zero and obtain x = ³√(16 – 2x)
[M1] Use iteration process correctly at least once
[A1] Obtain final answer 2.256
[A1] Show sufficient iterations to 6 sf to justify answer or show a sign change in the interval (2.2555, 2.2565)

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