where \(a_{k}\) are constants and \(a_{n} \neq 0\) is called the leading coefficient. The degree of the polynomial is \(n\text{.}\) Polynomials are quite important in numerical analysis.
A power function has form \(f(x) = x^{n}\) for \(n\) a nonnegative integer. These functions are the building blocks of polynomials. The graph of \(f(x) = x^{n}\) for \(n=0,1,2,3,4,5,6\text{:}\)
The 3 functions for \(n=2,4,6\) above are solid and shaped like cups opening up with the higher power growing faster with \(|x|>1\) a flatter bottom inside the interval \([-1,1]\text{.}\) The odd powers are dashed with a similar features in that the higher powers are steeper outside of the interval \([-1,1]\) and flatter inside the interval.
This is a simple compact form of a polynomial and this can be useful for various types of theorems and it easy to see the degree and leading coefficient of the polynomial.
where the roots are \(x_{0},x_{1},x_{m}\) and have multiplicity\(k_{0}, k_{1}, \ldots k_{m}\) respectively with \(a=a_{n}\) from the standard form above, called the leading coefficient and the multiplicities satisfy: \(n = k_{0}+k_{1} + \cdots + k_{m}\text{,}\) where \(n\) is the degree. Not all polynomials in standard form can be written in factored form, which assumes that all roots are real.
The factored form obviously gives you the roots directly, however, one can easily create a graph from this form as well. The behavior as \(|x| \rightarrow \infty\text{,}\) is the same as the power function with leading coefficient. That is it behaves like \(a_{n} x^{n}\text{.}\)
it is in Horner’s Form, which is generally an efficient way of evaluating the polynomial at a point, both in terms of efficiency of operations and in terms of rounding errors as will be seen in Section 2.3.
The multiplicity of the root \(x_i\) is denoted \(k_i\text{,}\) which is the power of the term \(x-x_i\) in the factored form. We will see in Section 3.1 an alternative way to find the multiplicity including the multiplicity of roots for functions other than polynomials.
The end behavior of the graph (shape of the curve for \(|x| \rightarrow \infty\)) is similar to the power function \(ax^{n}\text{,}\) where \(n\) is the degree and \(a\) is the leading coefficient.
The intervals of positive and negative values of the polynomial can be found by first understanding the sign of the power function \(ax^{n}\) for \(x>0\) and \(x<0\text{,}\) which depends on the sign of \(a\) and if \(n\) is even or odd. Once the sign of \(p\) is known for large \(x\) (in the positive or negative direction), we know that \(p\) will only switch sign over a zero with odd multiplicity.
The power function is \(y=3x^{4}\text{,}\) where the 3 arises from the coefficient in front of the terms in (1.1.4) amd the 4 is the degree of the polynomial, the sum of all of the multiplicities.
Since the power function for large \(|x|\) is \(3x^{4}\text{,}\) left of \(-1\) and right of \(2\) the function is positive. Recalling that the polynomial changes sign over roots of odd multiplicity however doesn’t over even multiplicity, one gets the following sign chart:
This polynomial is in factored form. The roots are \(2, -1\) and \(-3\) with multiplicities 3, 2 and 1 respectively. Also, the degree of the polynomial is the sum of the multiplicities or 6. Notice that the term in front is \(-1/5\text{,}\) so the leading term is \(-\frac{1}{5}x^{6}\text{.}\)
The intervals on which the polynomial is positive and negative is found the following way. Note that the leading term is \(-\frac{1}{5}x^{6}\text{,}\) the far behavior of \(p\) is the same as this so it is negative for \(|x| \rightarrow \infty\text{.}\) The polynomial will change sign over the roots of odd multiplicity or at \(x=2\) and \(x=-3\text{.}\) Therefore the sign of the polynomial can be summarized in the following sign chart:
Also as we will see in Section 6.4, some polynomials have properties that seem to be nice may not be so nice. For example a polynomial with equally spaced roots have a nice form, but the local extrema get very large. The following example is a polynomial with equally-spaced roots.
A Wilkinson polynomial is a polynomial of some degree \(n\) (an integer) with the following property: the polynomial as roots at \(1,2,3, \ldots, n\text{,}\) each of which has multiplicity 1. Write down the Wilkinson polynomial of degree 5.
In Chapter 4, we will cover how to find a polynomial that passes through a given set of points. The next example is a fundamental building block of such polynomials called a Lagrange polynomial.
Since there are 3 points that it must satisfy, unless there is a special relationship between \(x_{0}, x_{1}\) and \(x_{2}\text{,}\) we will probably need a quadratic. We know two of the zeros of the polynomial, so a form is:
One of the most important analytical aspects of polynomials is the Fundamental Theorem of Algebra, which states that the number of roots of a polynomial equals the degree.
Let \(P(x)\) be a non-constant single-variable polynomial with complex roots and degree \(n\text{.}\) Then \(P\) has exactly \(n\) roots if each root is counted relative to its multiplicity.
Although the statement is quite simple, the proof is not as it generally requires significant knowledge of Complex Analysis as is generally shown at the end of the course. See Complex Analysis by Howell and Mathews as a nice open-source text. 1
Another important aspect of polynomials is that if there is division between two polynomials and this is often seen in polynomials long division. Consider the following example:
Let \(a(x)\) and \(b(x)\) be polynomials where the degree of \(b\) is smaller than that of \(a\text{.}\) Then there exists polynomials \(q(x)\) and \(r(x)\) such that
