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Section7.5Polynomial Functions

Subsection7.5.1Polynomials Defined

One of the most important mathematical families of functions is the collection of polynomials. Given the independent variable \(x\), a polynomials is a calculation that only includes addition and multiplication of real numbers and the value of \(x\). Repeated multiplication by \(x\) corresponds to positive integer powers of \(x\).

Definition7.5.1
A function \(p(x)\) is a polynomial if it can be written \begin{equation*} p(x) = a_0 + a_1 \cdot x + a_2 \cdot x^2 + \ldots + a_n x^n \end{equation*} (but traditionally written in decreasing order of powers) for some integer \(n \ge 0\) and real numbers \(a_0, a_1, \ldots, a_n\). The value of \(n\) is called the degree of the polynomial and the constants \(a_0, a_1, \ldots, a_n\) with \(a_n \ne 0\) are called the coefficients of the powers \(x^0, x^1, \ldots, x^n\). The value \(a_n\) is called the leading coefficient.
Example7.5.2
Determine the coefficients and degree of each polynomial.
  1. \(p(x) = x^2+3x-5\)
  2. \(f(x) = 3x(x-5)(2x+1)\)
Solution

Subsection7.5.2Polynomials in Factored Form

Although polynomials are defined as a sum of whole number power functions, we frequently need to think about them as products. This is based on an important mathematical principle of solving equations, that a product can equal zero if and only if one of the factors is equal to zero.

Polynomials are frequently used to model relationships between variables where it is know that the dependent variable is zero for certain values of the independent variable. A model can be formed by multiplying together factors that individually are zero at each of the desired locations. An expression of the form \(x-c\) is often used because \(x-c\) equals zero when \(x=c\). A product of the form \((x-a)(x-b)(x-c)\) will equal zero at \(x=a\) (from the first factor), at \(x=b\) (from the second factor), and at \(x=c\) (from the third factor).

Example7.5.4
For each set of values, find a function that has zeros at each value in the set.
  1. \(\{0, 4\}\)
  2. \(\{-2, 3\}\)
  3. \(\{-1, 1, 3\}\)
Solution

In the previous example, because we only used simple factors, the leading coefficient was always 1. Requiring that the function also passes through one additional, non-zero point corresponds to a vertical rescaling by multiplying by some number. Also, in modeling settings, it is often better to leave the formula in factored form.

Example7.5.5
Using the same set of zeros as in the previous example, find a function that has the specified zeros and passes through the given point.
  1. Zeros at \(\{0, 4\}\) and \(f(1)=2\)
  2. Zeros at \(\{-2, 3\}\) and \(f(0)=3\)
  3. Zeros at \(\{-1, 1, 3\}\) and \(f(2)=-6\)
Solution

In the examples thus far, the function has crossed the axis at each of the zeros. This is a consequence of using a factor that behaves like a linear function. If we choose factors that include powers of the basic factor, we will functions that behave like the corresponding power functions near the zeros. The power with which a factor is repeated is called the multiplicity of the zero.

Example7.5.6
Find the equation of a function that touches, but does not cross, the \(x\)-axis at \(x=-2\) and at \(x=3\), and crosses the axis at \(x=0\), similar to the graph shown below, with a value \(f(1)=5\).

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Solution

Subsection7.5.3Factoring Polynomials

Just as every positive integer has a unique prime factorization, every polynomial can be factored in a unique way (up to order of factors). In the history of mathematics, this result is actually fairly remarkable and is closely related to the development of complex numbers. The Fundamental Theorem of Algebra guarantees that every polynomial of degree \(n\) has exactly \(n\) zeros, counted by multiplicity. The zeros may occur at complex numbers, and if so, they will occur in complex conjugate pairs.

In addition to the Fundamental Theorem of Algebra, the Factor–Root theorem connects the zeros of the polynomial to the factors.

Zeros that are complex conjugate pairs can be written using simple factors, but it is more common to include a single quadratic factor so that we only need to deal with real numbers. A basic quadratic has a leading coefficient of \(a=1\), and so is of the form \(x^2+bx+c\). The quadratic formula states that the zeros of such a quadratic are at \begin{equation*} x = \frac{-b \pm \sqrt{b^2-4c}}{2}. \end{equation*} The zeros will be complex conjugate pairs if \(b^2-4c \lt 0\). When a quadratic has complex conjugate zeros, we say the quadratic is irreducible because it doesn't factor with real factors.

The take-home message of all this is that for any polynomial that you start with, it is (in principle) possible to factor the polynomial completely so that every real zero corresponds to factors of the form \((x-c)\) (where \(c\) is the value of the zero) and all remaining factors are irreducible quadratics.

In general, factoring a polynomial is very difficult. But there are some special cases that are relatively easy and which are typically used in practice problems. Otherwise, we need to rely on computational tools to answer such questions. Your task is to know the special cases where the work is manageable. For more details, refer to the appendix on polynomial factoring.

Subsection7.5.4End Behavior (Limits) of Polynomials

The factors of a polynomial tell us where and how many times the graph crosses the \(x\)-axis. If there is a value \(x\) that makes a factor equal zero, then that value is an \(x\)-intercept. If the factor is repeated an even number of times, then although the graph comes and touches the axis, the graph will not cross. It is as though it bounces off the axis. So knowing the factors allows us to determine much of the shape of the graph.

All polynomials (except constant functions) have ends that go to infinity. We refer to the left side of a graph as \(x \to -\infty\) (\(x\) moves along the axis to the left towards infinity) and the right side of the graph as \(x \to +\infty\) (\(x\) moves along the axis to the right towards infinity). If the graph goes up forever then we say \(y \to +\infty\) and if the graph goes down forever then we say \(y \to -\infty\).

Example7.5.8
Describe the end behavior of the polynomial \(y=x^2\).
Solution
Example7.5.9
Describe the end behavior of the polynomial \(y=x^3\).
Solution

The degree and leading coefficient of a polynomial completely determine the end behavior. If the degree is even, then the polynomial will behave like a parabola with both ends ultimately going in the same direction (both up or both down). However, if the degree is odd, then the polynomial will behave like a cubic where the ends ultimately go in opposite directions (one up and one down). The sign of the leading coefficient determines what happens as \(x \to +\infty\), with the graph going up if the leading coefficient is positive and going down if the coefficient is negative. Together, this information completely determines the end behavior.

Example7.5.10
Describe the end behavior of the polynomial \(f(x) = -2x^4+x^3+5x^2-5\).
Solution

When a polynomial is written in factored form, you only need to multiply enough to determine the degree (or just decide if even/odd) and the sign of the leading coefficient.

Example7.5.11
Describe the end behavior of the polynomial \(f(x) = -3x^2(x-4)(x+1)^2\).
Solution