Quartic roots and an Argand triangle

A, B and C represent complex numbers a=0,b=3,c=−2+ia=0,b=3,c=-2+i. These numbers are roots of f(z)=0f(z)=0, where f(z)f(z) is a quartic polynomial with real coefficients and zz is complex.

(a) Express f(z)f(z) as a product of two quadratic factors with real coefficients. marks

(b) Sketch an Argand diagram showing all roots. marks

(c) W represents ww, with c=iwc=iw. Find AC→⋅AW→\overrightarrow{AC}\cdot\overrightarrow{AW} and the area of triangle APW, where P represents c−bc-b. marks

[2024/DHS/Prelim/I/7]

Total marks: 9

Solution

(a)

Real quadratic factors

A,B,CA,B,C represent a=0a=0, b=3b=3, c=−2+ic=-2+i, roots of a quartic f(z)=0f(z)=0 with real coefficients. Express f(z)f(z) as a product of two real quadratic factors.

Hint

  1. Find the fourth root by conjugation.

  2. Pair the roots into real quadratics.

Answer · (a)

f(z)=K(z2−3z)(z2+4z+5)f(z)=K(z^2-3z)(z^2+4z+5), K∈R∖{0}K\in\mathbb R\setminus\{0\}.

Printed K=1 assumes an unstated monic polynomial.

(b)

Plot the roots

Sketch an Argand diagram showing all the roots.

Explore this idea

Hint

  1. Plot (0,0)(0,0), (3,0)(3,0), (−2,1)(-2,1) and (−2,−1)(-2,-1).

Quartic roots on an Argand diagram
Quartic roots on an Argand diagramScroll across to see the full diagram.

Answer · (b)

Roots at (0,0),(3,0),(−2,1),(−2,−1)(0,0),(3,0),(-2,1),(-2,-1).

Explore the idea

Roots of the quartic

Drag to look around the Argand plane. Real parts run along the horizontal axis and imaginary parts along the vertical axis.

Reset the controls
Key idea

Real roots lie on the axis; non-real roots come as a mirror pair about it.

Ideas to try
The dashed segment joining the conjugate pair is vertical and crosses the real axis at −2-2.
(c)

Quarter-turn, dot product and area

WW represents ww, where c=iwc=iw, and PP represents c−bc-b. Find AC→⋅AW→\overrightarrow{AC}\cdot\overrightarrow{AW} and the area of triangle APWAPW.

Explore this idea

Hint

  1. Solve w=c/i=−icw=c/i=-ic.

  2. Write the vectors in components and use the dot product and the 2×22\times2 determinant.

Answer · (c)

AC⋅AW=0AC\cdot AW=0; area 11/211/2.

Explore the idea

A quarter-turn makes a right angle

Turn CC about the origin. The dot product AC→⋅AW→\overrightarrow{AC}\cdot\overrightarrow{AW} is zero exactly when the turn is 90∘90^\circ.

Clockwise turn of C90°
Try an example
W
1 + 2i
AC · AW
0
area of APW
5.5
  • Moves as the slider changes; a faint path shows where it goes
Key idea

Multiplying by −i-i rotates a vector through 90∘90^\circ clockwise, so ACAC and AWAW are perpendicular and have the same length 5\sqrt5.

Ideas to try
  1. Slide to 90∘90^\circ: the dot product is 00 and the area is 5.55.5.
  2. Slide back toward 0∘0^\circ and watch the dot product grow to ∣C∣2=5|C|^2=5.
Takeaway

Roots first, geometry second

Conjugation completes the root set, and multiplication by ±i\pm i turns vectors through right angles, which makes dot products vanish.