Applied Algebra: Modeling and functions

Subsection Classifying Polynomials by Degree

The graph of a polynomial function depends first of all on its degree. We have already studied the graphs of polynomials of degrees \(0\text{,}\) \(1\text{,}\) and \(2\text{.}\)

• A polynomial of degree \(0\) is a constant, and its graph is a horizontal line. An example of such a polynomial function is \(f(x) = 3\text{.}\)

• A polynomial of degree \(1\) is a linear function, and its graph is a straight line. The function \(f(x) = 2x - 3\) is an example of a polynomial of degree \(1\text{.}\)

  

• Quadratic functions, such as \(f(x) = -2x^2 + 6x + 8\text{,}\) are polynomials of degree \(2\text{.}\) The graph of every quadratic function is a parabola, with the same basic shape as the standard parabola, \(y = x^2\text{.}\) It has one turning point, where the graph changes from increasing to decreasing or vice versa. The turning point of a parabola is the same as its vertex.

  

Subsection Cubic Polynomials

Do the graphs of all cubic, or third-degree, polynomials have a basic shape in common? We can graph a few examples and find out. Unlike the basic parabola, the graph of \(y = x^3\) is always increasing. At the origin, however, it changes from concave down to concave up. A point where the graph changes concavity is called an inflection point.

Graph the cubic polynomial \(P(x) = x^3 - 4x\) and compare its graph with that of the basic cubic, \(y = x^3\text{.}\)

Solution

The graph of the basic cubic is shown in figure (a) below. To help us understand the graph of the polynomial \(P(x) = x^3 - 4x\text{,}\) we will evaluate the function to make a table of values. We can do this by hand or use the Table feature on the graphing calculator.

\(x\)	\(-3\)	\(-2\)	\(-1\)	\(0\)	\(1\)	\(2\)	\(3\)
\(P(x)\)	\(-15\)	\(0\)	\(3\)	\(0\)	\(-3\)	\(0\)	\(15\)

The graph of \(P(x) = x^3 - 4x\) is shown in figure (b). It is not exactly the same shape as the basic cubic—it has two turning points—but it is similar, especially at the edges of the graphs.

Despite the differences in the central portions of the two graphs, they exhibit similar long-term behavior.

• For very large and very small values of \(x\text{,}\) both graphs look like the power function \(y = x^3\text{.}\)
• The \(y\)-values increase from \(-\infty\) toward zero in the third quadrant, and they increase from zero toward \(+\infty\) in the first quadrant. Or we might say that the graphs start at the lower left and extend to the upper right.

All cubic polynomials display this behavior when their lead coefficients (the coefficient of the \(x^3\) term) are positive.

• Both of the graphs in Example 7.13 are smooth curves without any breaks or holes. This smoothness is a feature of the graphs of all polynomial functions.
• The domain of any polynomial function is the entire set of real numbers.

Subsection Quartic Polynomials

Now let’s compare the long-term behavior of two quartic, or fourth-degree, polynomials.

Graph the polynomials \(f(x) = x^4 - 10x^2 + 9\) and \(g(x) = x^4 + 2x^3\text{,}\) and compare.

Solution

For each function we make a table of values.

\(x\)	\(-4\)	\(-3\)	\(-2\)	\(-1\)	\(0\)	\(1\)	\(2\)	\(3\)	\(4\)
\(f(x)\)	\(105\)	\(0\)	\(-15\)	\(0\)	\(9\)	\(0\)	\(-15\)	\(0\)	\(105\)
\(G(x)\)	\(128\)	\(27\)	\(0\)	\(-1\)	\(0\)	\(3\)	\(32\)	\(135\)	\(384\)

The graphs are shown below. All the essential features of the graphs are shown in these viewing windows. The graphs continue forever in the directions indicated, without any additional twists or turns. You can see that the graph of \(f\) has three turning points, and the graph of \(g\) has one turning point.

As in Example 7.13, both graphs have similar long-term behavior. The \(y\)-values decrease from \(-\infty\) toward zero as \(x\) increases from \(-\infty\text{,}\) and the \(y\)-values increase toward \(+\infty\) as \(x\) increases to \(+\infty\text{.}\) This long-term behavior is similar to that of the power function \(y = x^4\text{.}\) Its graph also starts at the upper left and extends to the upper right.

In Examples 7.13 and 7.15, we have seen polynomials of degree 3 and degree 4, whose graphs are illustrated in the next box. In the Homework Problems, you will consider more graphs to help you verify the following observations.

Long-Term Behavior of Polynomial Functions.

1. A polynomial of odd degree (with positive lead coefficient) has negative \(y\)-values for large negative \(x\) and positive \(y\)-values for large positive \(x\text{.}\)

2. A polynomial of even degree (with positive lead coefficient) has positive \(y\)-values for both large positive and large negative \(x\text{.}\)

Subsection \(x\)-Intercepts and the Factor Theorem

In Chapter 6, we saw that the x-intercepts of a quadratic polynomial, \(f(x) = ax^2 + bx + c\text{,}\) occur at values of \(x\) for which \(f(x) = 0\text{,}\) that is, at the real-valued solutions of the equation \(ax^2 + bx + c = 0\text{.}\) The same holds true for polynomials of higher degree.

Solutions of the equation \(P(x) = 0\) are called zeros of the polynomial \(P\text{.}\) In Example 7.13, we graphed the cubic polynomial \(P(x) = x^3 - 4x\text{.}\) Its \(x\)-intercepts are the solutions of the equation \(x^3 - 4x = 0\text{,}\) which we can solve by factoring the polynomial \(P(x)\text{.}\)

The zeros of \(P\) are \(0\text{,}\) \(2\text{,}\) and \(-2\text{.}\) Each zero of \(P\) corresponds to a factor of \(P(x)\text{.}\) This result suggests the following theorem, which holds for any polynomial \(P\text{.}\)

The factor theorem follows from the division algorithm for polynomials. We will consider both of these results in more detail in the Homework problems.

Because a polynomial function of degree \(n\) can have at most \(n\) linear factors of the form \((x - a)\text{,}\) it follows that \(P\) can have at most \(n\) distinct zeros.

Another way of saying this is that if \(P(x)\) is a polynomial of \(n\)th degree, the equation \(P(x) = 0\) can have at most \(n\) distinct solutions, some of which may be complex numbers. (We consider complex numbers in Section 7.3.)

Because only real-valued solutions appear on the graph as \(x\)-intercepts, we have the following corollary to the factor theorem.

If some of the zeros of \(P\) are complex numbers, they will not appear on the graph, so a polynomial of degree \(n\) may have fewer than \(n\) \(x\)-intercepts.

Subsection Zeros of Multiplicity Two or Three

The appearance of the graph near an \(x\)-intercept is determined by the multiplicity of the zero there.

• Both real zeros of the polynomial \(g(x) = x^4 - 3x^2 - 4\) in Example 7.18b are of multiplicity one, and the graph crosses the \(x\)-axis at each intercept.
• The polynomial \(f(x) = x^3 + 6x^2 + 9x\) in Example 7.18a has a zero of multiplicity two at \(x = -3\text{.}\) The graph of \(f\) just touches the \(x\)-axis and then reverses direction without crossing the axis.

To understand what happens in general, compare the graphs of the three polynomials below.

1. In figure (a), \(L(x) = x - 2\) has a zero of multiplicity one at \(x = 2\text{,}\) and its graph crosses the \(x\)-axis there.

2. In figure (b), \(Q(x) = (x - 2)^2\) has a zero of multiplicity two at \(x = 2\text{,}\) and its graph touches the \(x\)-axis there but changes direction without crossing.

3. In figure (c), \(C(x) = (x - 2)^3\) has a zero of multiplicity three at \(x = 2\text{.}\) In this case, the graph makes an S-shaped curve at the intercept, like the graph of \(y = x^3\text{.}\)

Near its \(x\)-intercepts, the graph of a polynomial takes one of the characteristic shapes illustrated above.

Graph the polynomial

\begin{equation*} f(x) = (x + 2)^3(x - 1)(x - 3)^2 \end{equation*}

Solution

The polynomial has degree six, an even number, so its graph starts at the upper left and extends to the upper right. Its \(y\)-intercept is

\begin{equation*} f(0) = (2)^3(-1)(-3)^2 = -72 \end{equation*}

\(f\) has a zero of multiplicity three at \(x = -2\text{,}\) a zero of multiplicity one at \(x = 1\text{,}\) and a zero of multiplicity two at \(x = 3\text{.}\) The graph has an S-shaped curve at \(x = -2\text{,}\) crosses the \(x\)-axis at \(x = 1\text{,}\) touches the \(x\)-axis at \(x = 3\text{,}\) and then changes direction, as shown below.

Sketch a rough graph of \(f(x) = (x + 3)(x - 1)^2\) by hand. Label the \(x\)- and \(y\)-intercepts.

Answer

Subsection Section Summary