FP1 June 2014 Q3 – Westie's Workshop

FP1 June 2014 Q3

EdexcelOld spec8 marksComplex Numbers Using Roots of Polynomials

3. Given that 2 and \(1 - 5\mathrm{i}\) are roots of the equation \[x^3 + px^2 + 30x + q = 0, \qquad p, q \in \mathbb{R}\]

(a) write down the third root of the equation. (1)

(b) Find the value of \(p\) and the value of \(q\). (5)

(c) Show the three roots of this equation on a single Argand diagram. (2)

Mark scheme (b)

Scheme	Marks
\(\big((x - (1 + 5i))(x - (1 - 5i))\big) = x^2 - 2x + 26\) \(\big((x - 2)(x - (1 \pm 5i))\big) = x^2 - (3 \pm 5i)x + 2(1 \pm 5i)\) M1: 1. Attempt to expand or 2. Use sum and product of the complex roots. A1: Correct expression	M1A1
\((x^2 - 2x + 26)(x - 2) = x^3 + px^2 + 30x + q\) Uses their third factor with their quadratic with at least 4 terms in the expansion	M1
\(p = -4, \qquad q = -52\) May be seen in cubic	A1, A1
OR
\(\mathrm{f}(1 + 5\mathrm{i}) = 0\) or \(\mathrm{f}(1 - 5\mathrm{i}) = 0\) Substitute one complex root to achieve 2 equations in \(p\) and / or \(q\)	M1
\(q - 24p - 44 = 0\) and \(10p + 40 = 0\) Both equations correct oe	A1
Solving for \(p\) and \(q\)	M1
\(p = -4, \quad q = -52\) May be seen in cubic	A1, A1
	(5)

Argand diagram with origin O: points 1 + 5i at (1, 5) and 1 - 5i at (1, -5), and the point 2 on the positive real axis; 1 and 2 marked on the real axis, 5 and -5 on the imaginary axis — Mark scheme (c)

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