Factor a real quartic by a conjugate pair

Do not use a calculator in answering this question.

(a) f(x)f(x) is a degree-4 polynomial with real coefficients. Explain whether f(x)=0f(x)=0 can have 3 non-real roots and 1 real root. mark

(b) One root of 2x4−15x3+ax2−63x+b=02x^4-15x^3+ax^2-63x+b=0, where a,ba,b are real, is 3−2i3-2i. Find the other roots and a,ba,b. marks

[2024/YIJC/Prelim/I/3]

Total marks: 7

Solution

(a)

Count non-real roots

f(x)f(x) is a degree-4 polynomial with real coefficients. Explain whether f(x)=0f(x)=0 can have 33 non-real roots and 11 real root.

Hint

  1. How do non-real roots of real polynomials pair up?

Answer · (a)

No: nonreal roots of real polynomials occur in conjugate pairs, counting multiplicity.

(b)

Quadratic factor from a conjugate pair

One root of 2x4−15x3+ax2−63x+b=02x^4-15x^3+ax^2-63x+b=0, with a,ba,b real, is 3−2i3-2i. Find the other roots and a,ba,b.

Explore this idea

Hint

  1. Write the real quadratic factor from the pair 3±2i3\pm2i.

  2. Factor the quartic as (quadratic)(quadratic) and compare coefficients.

Answer · (b)

a=48,b=52a=48,b=52; other roots 3+2i,(3+i23)/4,(3−i23)/43+2i,(3+i\sqrt{23})/4,(3-i\sqrt{23})/4.

Explore the idea

Two conjugate pairs

All four roots are non-real, arranged as two pairs of mirror images in the real axis.

Reset the controls
Key idea

A real quartic with no real roots has its four roots in two mirror-image pairs, one real quadratic factor for each.

Ideas to try
  1. The dashed segments are vertical and cross the real axis at the real part of each pair: 33 and 34\frac34.
  2. Check that the pair 3±2i3\pm2i has sum 66 and the other pair has sum 32\frac32. The four roots add to 152\frac{15}2.
Takeaway

Pairs make real factors

Each conjugate pair p±qip\pm qi contributes a real factor x2−2px+p2+q2x^2-2px+p^2+q^2. Match coefficients to find the rest.