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Read cubic roots and rotate roots of a complex quadratic

Do not use a graphic calculator in answering this question.

(a) The diagram shows y=f(x)y=f(x) for a cubic polynomial with real coefficients. What can be said about all roots of f(x)=0f(x)=0? marks

(b) The equation 2z2−(7+6i)z+11+ic=02z^2-(7+6i)z+11+ic=0, where cc is a nonzero real number, has a root z=3+4iz=3+4i. Show that c=−2c=-2 and determine the other root in Cartesian form. Hence find the roots of 2w2+(−6+7i)w−11+2i=02w^2+(-6+7i)w-11+2i=0. marks

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Qualitative cubic sketch with an increasing right-hand end and decreasing left-hand end. It crosses the x-axis once on the negative side, has a local maximum on the y-axis above the x-axis, then a local minimum to the right that also stays above the x-axis. Neither turning point touches the axis. The axes have no numerical scales.
Qualitative cubic sketch with an increasing right-hand end and decreasing left-hand end. It crosses the x-axis once on the negative side, has a local maximum on the y-axis above the x-axis, then a local minimum to the right that also stays above the x-axis. Neither turning point touches the axis. The axes have no numerical scales.
Qualitative cubic sketch with an increasing right-hand end and decreasing left-hand end. It crosses the x-axis once on the negative side, has a local maximum on the y-axis above the x-axis, then a local minimum to the right that also stays above the x-axis. Neither turning point touches the axis. The axes have no numerical scales.

[2017/DHS/I/8(a),(b)]

CJC · Review problems · Question 8

Separate the real-coefficient cubic in(a) from the complex-coefficient quadratic in(b). Use the sum or product of roots for the second quadratic root.

Read the worked solution