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MTH 441
Worksheet 03: Solution of the Cubic
Fall 2016
The formula for finding the roots of some cubic polynomials ! f (x) = x 3 + bx 2 + cx + d was discovered in about
1515 by Scipione del Ferro and extended to all cubic polynomials by Fontana (Tartaglia) and Cardano by about
1545. At first, its discovery was keep secret because mathematicians were highly competitive at that time. Once
the cubic formula became known, it dramatically changed mathematics by forcing mathematicians to accept the
Complex Numbers as a nessary extension of the Real Numbers! Today, complex numbers are viewed as an
essential part of mathematics: Beyond modern algebra, they are used in algebraic geometry, number theory,
compex analysis, complex geometry, numerical anaysis, applied mathematics and mathematical physics.
Here is an outline of the once secret method for solving cubics:
1. Let ! f (x) = x 3 + bx 2 + cx + d where b, c, and d are constants. Show that the substitution ! x = X − b 3 will
transform ! f (x) into a cubic of the form ! g(X) = X 3 + pX + q where p and q are expressions in b, c, and d.
What are p and q in terms of b, c, and d?
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2. Let A and B be variables. Show that the identity ! ( A + B ) − 3AB ( A + B ) − A 3 + B 3 = 0 is valid.
3
3. Let ! X = A + B as in Problem 2. Express the identity of Problem 2 and in the form of the simplified cubic
polynomial ! g ( X ) of Problem 1 and find formulas for ! A 3 + B 3 and A 3 B 3 in terms of p and q.
4. a) By using subsititution, show that ! A 3 and ! B 3 are roots of the quadratic equation
! Z 2 − A3 + B3 Z + A3B3 = 0 .
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)
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b) Using your results of Problem 3, rewrite the ! Z 2 − A 3 + B 3 Z + A 3 B 3 = 0 in terms of p and q
5. Using the quadratic formula, find the two solutions ! Z1 and ! Z 2 of the quadratic equation found in Problem
4b for Z. Your two solutions should be expressed in terms of p and q.
6. Use Problem 4a and Problem 5 to find formulas for A and B in terms of p and q.
7. Let ! ω =
−1+
( 3 ) i , show that ! ω
2
2
=
−1−
( 3 )i , !ω
2
3
= 1 , and ! ω 2 + ω + 1 = 0 .
8. Assume ! r1 = A + B, r2 = ω A + ω 2 B, r2 = ω 2 A + ω B . Using your answers for Problems 1-7, show that
! g ( r1 ) = g ( r2 ) = g ( r3 ) = 0 where ! g(X) = X 3 + pX + q .
9. Use the ideas and results of Problems 1-8 to find all the roots of
a) ! g(u) = u 3 − 3u + 2
b) ! g(u) = u 3 + 3u − 2 .
As discussed in class and in Appendix 5, you may need to use the polar form of a complex number to
find all of its roots.