Complex numbers: Complex polynomials
The notion of a complex polynomial
Solving equations is an important activity in mathematics. With complex numbers we can deal with more equations than with some real numbers. A common type is the polynomial equation. In this section we discuss the concept of a complex polynomial and polynomial equations. They are extensions of the concepts for all real numbers as treadted in the theory of the concept polynomial.
An expression of the form
\[a_nz^n+a_{n-1} z^{n-1}+ \cdots + a_1z+a_0\] in which #a_0,\ldots,a_n# are complex numbers and #z# is a variable, is called a complex polynomial in #z#. If #a_n\neq 0#, then #n# is called the degree of the polynomial. A polynomial of the #1# is also called linear and a polynomial of degree #2# is called quadratic. The numbers #a_0,\ldots,a_n# are called the coefficients of the polynomial. If they are real, then we call the polynomial real. The number #a_0# is called the constant term, the number #a_n# is called the leading coefficient.
Let #p(z)# be a polynomial. The equation #p(z)=0#, and each equation that has the same form by reducation after moving all terms to the left hand side, is called a polynomial equation. If #p(w) = 0#, then #w# is called the zero of the polynomial, or sometimes a root or solution of the polynomial equation #p(z) = 0#.
The image that adds the value #p(z)# to a complex number #z#, is called a polynomial function.
The polynomial #z^3-\ii# is of degree #3#. The polynomial #z^2+z+1# is real and quadratic.
The definition does not add a degree to the polynomial #0#. Sometimes one agrees that the degree of #0# is equal to #-\infty#. For the rules of calculation, it is important that the degree of #0# is smaller than #0#, the degree of a constant polynomial that is not equal to #0#.
The polynomial #\ii# has degree #0#. The corresponding polynomial is the constant function #C_{\ii}#, which adds the number #\ii# to #z#. Again, we use the notation #C_a#, although this was a real function when introducing the constant function. The domain of #C_a# is expanded to #\mathbb{C}#. The function #C_a# with domain #\mathbb C# is real valued if and only if #a# is real.
The following lines show how many terms can add, subtract and multiply.
Arithmetic operations for complex polynomials
Let #f(z)=a_mz^m+a_{m-1}z^{m-1}+\cdots+ a_0# and #g(z)=b_nz^n+b_{n-1}z^{n-1}+\cdots +b_0# be two polynomials of degree #m# respectively #n#. We assume that #m\ge n#. Also, let #c# be a real number.
The following expressions are also polynomials:
- #c\cdot f(z) = (c\cdot a_m)z^m+(c\cdot a_{m-1})z^{m-1}+\cdots+ (c\cdot a_0)#
- # f(z)+g(z) = (a_m+b_m)z^{m}+(a_{m-1}+b_{m-1})z^{m-1}+\cdots+ (a_0+b_0)#, in which #b_j=0# for #j\gt n#
- # f(z)\cdot g(z) = (a_m\cdot b_n)z^{m+n}+(a_{m-1}b_{n}+a_m\cdot b_{m-1})z^{m-1}+\cdots+ (a_0\cdot b_0)#, in which the coefficient of #z^k# for general #k# is equal to #\displaystyle \sum_{j=0}^{k}a_j\cdot b_{kj}#.
The quotient #\dfrac{f(z)}{g(z)}# of two polynomials is not always a polynomial, but does give a rational function. We deal with this later on. We can, however, divide #g(z)# on #f(z)# with remainder. Again, we will deal with this later on.
As a result of these laws, which correspond to what we know about real polynomials, all kinds of rules that we know about real polynomials, also hold for complex numbers. Examples include the banana formula, notable products , and the binomial of Newton .
- The degree of #c\cdot f(z)# is the degree of #f(z)# as #c\ne0#.
- The degree of #f(z)\cdot g(z)# is the sum of the degrees of #f(z)# and #g(z)#.
- If #m\gt n#, then the degree of #f(z)+g(z)# is equal to the degree of #f(z)#.
- If #m=n#, then the degree of #f(z)+g(z)# is less than or equal to the degree of #f(z)#.
To prove the statements we write #f(z)=a_mz^m+a_{m-1}z^{m-1}+\cdots+ a_0# and #g(z)=b_nz^n+b_{n-1}z^{n-1}+\cdots +b_0# as above. We assume that #a_m# and #a_n# are not #0#, and that #m\ge n#.
1. The leading coefficient of #c\cdot f(z)# is #c\cdot a_m#; it occurs as a coefficient of #z^m#. Therefore, the degree of #c\cdot f(z)# is equal to #m#.
2. The leading coefficient of #f(z)\cdot g(z)# is #a_m\cdot b_n#; it occurs as a coefficient of #z^{m+n}#. Therefore, the degree of #f(z)\cdot g(z)# is equal to #m+n#.
3. The leading coefficient of #f(z)+g(z)# is #a_m#; it occurs as a coefficient of #z^{m}#. Therefore, the degree of #f(z)+g(z)# is equal to #m#.
4. The leading coefficient of #f(z)+g(z)# is #a_m+b_m#, unless this number is equal to zero; it occurs as a coefficient of #z^{m}#. Therefore, the degree of #f(z)+g(z)# is less than or equal to #m#.
This value is obtained by entering in #-3+3\cdot \ii# for #z# in #z^2+2\cdot \ii\cdot z+3#. This gives
\[ \begin{array}{rcl}\left(-3+3\cdot \ii\right)^2+2\cdot \ii\cdot \left(-3+3\cdot \ii\right)+3&=&-18\cdot \ii-6\cdot \ii-6+3\\&=&-3-24\cdot \ii
\end {array}\]
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