DPST1013 Chap.3 Complex Numbers and Polynomial Roots
Complex Numbers and Polynomial Roots
Complex numbers are treated both as algebraic expressions and as points on the Argand plane. Cartesian form supports addition and conjugation, while polar form turns multiplication into scaling and rotation. De Moivre's theorem then produces powers, complete root sets and factorisations of complex and real polynomials.
What this chapter covers
- 01
Cartesian and polar form
- 02
Modulus and argument
- 03
Conjugates and division
- 04
De Moivre's theorem
- 05
Powers and complete root sets
- 06
Conjugate factors of real polynomials
Worked example · free
Find every cube root of a negative real number
- 1Write -8 as modulus 8 with argument pi plus full turns.
- 1Each root has modulus 2 and argument (pi + 2pi k)/3.
- 1Use k = 0, 1 and 2 to get angles pi/3, pi and 5pi/3.
- 1Convert to 1 + i sqrt(3), -2 and 1 - i sqrt(3).
Key terms
- Complex modulus
- The non-negative distance of a complex number from the origin.
- Principal argument
- A chosen representative angle for a non-zero complex number within a stated interval.
- Complex conjugate
- The reflection of a complex number across the real axis.
- De Moivre theorem
- A rule that raises a polar modulus to a power and multiplies its argument by that power.
Complex Numbers and Polynomial Roots FAQ
How do I choose the correct argument?
Plot the signs of the real and imaginary parts to identify the quadrant, then use an inverse tangent only for the reference angle. Adjust that angle into the requested principal interval before reporting the polar form.
Why are nth roots equally spaced?
Every representation of the target angle differs by a full turn. Dividing those angles by n creates increments of one full turn divided by n, so the roots form a regular polygon around the origin.
Why do non-real roots come in conjugate pairs?
For a polynomial with real coefficients, conjugating the equation preserves every coefficient. Therefore the conjugate of any non-real root also satisfies the polynomial. Pairing them produces a real quadratic factor.
When is polar form more useful than Cartesian form?
Polar form turns multiplication and division into operations on moduli and arguments, and it makes powers and roots systematic. Cartesian form remains more direct for addition, subtraction and comparison of real and imaginary parts. Convert deliberately rather than automatically.
After a polar calculation, use the expected quadrant and modulus to check the result before converting back to exact Cartesian coordinates if required.
How do I apply De Moivre's theorem without losing solutions?
For a power, multiply the argument and raise the modulus. For an nth-root problem, include every argument obtained by adding two pi times an integer before dividing by n. Select any n consecutive integer values to obtain the distinct roots. Listing the target number of solutions in advance prevents stopping after the principal root, and a regular-polygon sketch confirms their angular spacing.
What does complex conjugation do geometrically?
Conjugation reflects a point in the real axis: the real part is unchanged and the imaginary part reverses sign. Consequently the modulus remains the same while the argument changes sign, up to the chosen argument interval. Multiplying a non-zero number by its conjugate gives the positive real value equal to its squared modulus, which is why conjugates rationalise complex denominators cleanly.
How can a known polynomial root help factor the polynomial?
The factor theorem converts a root r into the factor z minus r. Divide the polynomial by that factor using long or synthetic division, then solve the lower-degree quotient. If the original coefficients are real and r is non-real, its conjugate supplies another factor immediately. Check the product of all factors against both the leading term and constant term before declaring the factorisation complete.
Why can the principal argument cause an apparent contradiction?
Arguments are defined only up to whole turns, whereas a principal argument is the single representative chosen from a specified interval. Two expressions may therefore have arguments differing by two pi while representing the same complex number.
State the interval, reduce the angle into it and avoid treating principal arguments as if they always add exactly under multiplication; a final wrap into the interval may be necessary.
Exam move
Choose the form that matches the operation. Use Cartesian form for addition, subtraction and real or imaginary parts. Use polar form for products, quotients, powers and roots. Sketch the quadrant before selecting an argument, list the expected number of roots before calculating, and pair conjugates when the polynomial has real coefficients. Verify one root by substitution and audit the spacing and modulus of the complete set.
Maintain a conversion routine that finds modulus, quadrant and argument separately, then reconstructs the Cartesian number as a check. When solving equations, distinguish equality of complex numbers from equality of moduli or arguments. Practise plotting complete root sets before simplifying exact coordinates; the geometry predicts symmetry and catches missing solutions.
For polynomial division, record the factor theorem statement first and multiply the final factors back to recover the original leading coefficient and constant term.
Working through Complex Numbers and Polynomial Roots in DPST1013? Sia is AskSia’s AI Mathematics tutor — ask any DPST1013 Complex Numbers and Polynomial Roots question and get a clear, step-by-step explanation grounded in how DPST1013 is taught and assessed. Read this chapter free, then take your hardest questions to Sia.