Complex numbers and polynomials - University College Cork

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A complex number z 0 is called a root or zero of a polynomial p if its value at z 0 iszero:p(z 0 ) = 0.A polynomial all of whose coefficients are real numbers may not have any real roots.The simplest example is the quadratic x 2 +1. This has two complex roots, viz., i, −i.Theorem 1. A polynomial of odd degree whose coefficients are real numbers, has atleast one real root.Intuitive solution. Consider their graphs in the plane. For instance, the graphs ofones of degree 1 are straight lines, which consist of sets of points of the plane thatlie above and below the horizontal axis, and contain a point of it. Likewise, graphsof cubics occupy regions of the plane that lie above and below the horizontal axis.Since these regions are joined (??), their union must contain a point of the real axis.Generally, if the degree ofp(x) = a n x n + · · · + a 0is odd, for all sufficiently large x > 0, the sign of the output p(x) matches that ofa n , while if x is large and negative, p(x) has the same sign as −a n . Thus, p takesboth positive and negative values, and being a smooth function it must thereforeassume the value zero. In other words, it must have a real root. □The reality of the coefficients specified in this theorem is essential. For example, thelinear polynomial x − i has no real root.Theorem 2 (Gauss). Every non-constant polynomial has at least one root, whichmay be complex. A polynomial of degree n has at most n distinct roots.This deep result is called the Fundamental Theorem of Algebra.granted.We take it forTheorem 3. Suppose p has real coefficients and z is one of its a complex roots.Then ¯z is also one of its roots.Proof. Supposep(x) = a 0 + a 1 x + a 2 x 2 + · · · + a n x n x ∈ C,where a 0 , a 1 , . . . , a n are real numbers. Then, by repeated application of Exercise 2,0 = p(z)= a 0 + a 1 z + a 2 z 2 + · · · + a n z n= a 0 + a 1 z + a 2 z 2 + · · · + a n z n= a 0 + a 1¯z + a 2¯z 2 + · · · + a n¯z n= a 0 + a 1¯z + a 2¯z 2 + · · · + ¯z n= p(¯z).Thus p(z) = 0 implies p(¯z) = 0. □Some further information about polynomials is given in an Appendix.4
3 Linear and quadratic polynomialsHere, we focus on linear and quadratic polynomials whose coefficients are real numbers,and restrict their domain of definition to be the set of real numbers. Wecan then consider their graphs and thence examine their properties in a geometricmanner. These are functions of the formwhere a, b, c are real, and a ≠ 0.3.1 Graphs of linesax + b, ax 2 + bx + c, x ∈ R,While the Greeks worked out the geometry of the straight line, they had no awayof representing them in an algebraic manner. It fell to Descartes to describe a wayof doing this: one introduces the cartesian plane R 2 of points with two coordinates,a procedure with which you are familiar. Graphs of functions are then identified assubsets of this plane whose intersection with every vertical line consists of at mosta single point. As a consequence, a circle is not the graph of a function, nor is theparabola {(x, y) ∈ R 2 : y 2 = 4x}.If c + mx is a linear polynomial, its graph in R 2 is the set of points{(x, y); y = mx + c, x ∈ R} = {(x, mx + c) : x ∈ R},which is abbreviated to y = mx + c; m is its slope, and c its y-intercept.Given two distinct points (x 1 , y 1 ), (x 2 , y 2 ) in its graph, so thaty 1 = mx 1 + c, y 2 = mx 2 + c,in which case x 2 ≠ x 1 , we can solve these equations for m, c and, doing so, find thatm = y 2 − y 1, c = y 1x 2 − y 2 x 1.x 2 − x 1 x 2 − x 1Substituting these into the equation, and rearranging the resulting expression, weobtain the familiar formula for the equation of a line through two points which arenot on the same vertical line:Equivalently in this case,y − y 1 = y 2 − y 1x 2 − x 1(x − x 1 ).(x 2 − x 1 )(y − y 1 ) = (y 2 − y 1 )(x − x 1 ).But this formula works even if x 2 = x 1 , provided that y 2 ≠ y 1 ; or if y 2 = y 1 andx 2 ≠ x 1 . But we can’t have y 2 − y 2 = x 2 − x 1 = 0 unless we wish to regard the entireplane as a line!To cover all possibilities, then, the equation of a line in R 2 is an expression of theformax + by + c = 0,where a 2 + b 2 > 0. This is the graph of a linear polynomial iff ab ≠ 0. It is thegraph of a constant polynomial if a = 0, b ≠ 0.5
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Complex numbers and polynomials - University College Cork

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