Polynomial Functions: Definition, Graphs, End Behavior, Zeros, and the Intermediate Value Theorem

3.2 Polynomial Functions

I. Definition of Polynomial Functions

• Definition 1: Let $n$ be a nonnegative integer and let $a<em>n, a</em>{n-1}, \dots, a<em>2, a</em>1, a<em>0$ be real numbers, with $a</em>n \neq 0$. A polynomial function is defined by the form $f(x) = a<em>n x^n + a</em>{n-1} x^{n-1} + \cdots + a<em>2 x^2 + a</em>1 x + a_0$. It is called a polynomial function of degree $n$, where $n$ is the highest power of the variable $x$ present in the function. The domain of all polynomial functions is all real numbers $(-\infty, \infty)$.

• The number $a_n$, which is the coefficient of the variable to the highest power, is called the leading coefficient. This coefficient is crucial for determining the end behavior of the polynomial's graph.

• Example 2 - Identifying Polynomial Functions:

  1. $p(x) = 5x^3 + \frac{1}{4}x^2 - 2$: This is a polynomial function of degree $3$ (highest power of $x$ is $3$) with a leading coefficient of $5$. All exponents are non-negative integers.

  2. $f(x) = x + 2 - 3x^3$: This is a polynomial function. When rearranged in standard form (descending powers of $x$), it is $-3x^3 + x + 2$. It has a degree of $3$ with a leading coefficient of $-3$.

  3. $g(x) = \sqrt{x}$: This is not a polynomial function because $x$ is raised to the power of $\frac{1}{2}$, which is not a nonnegative integer. For a function to be a polynomial, all exponents on the variables must be non-negative integers.

  4. $h(x) = \frac{1}{x+1}$: This is not a polynomial function because it involves an x in the denominator, meaning there is division by a variable expression. Polynomial functions do not have variables in the denominator.

  5. $F(x) = 3$: This is a constant function. It can be written as $3x^0$. Therefore, it is a polynomial function of degree $0$ with a leading coefficient of $3$. (Note: degree $0$ means the highest power of $x$ is $0$).

  6. $G(x) = -2x(x + 3)$ : This is a polynomial function. When expanded, it becomes $-2x^2 - 6x$. It has a degree of $2$ with a leading coefficient of $-2$. The highest power of $x$ is $2$.

II. Smooth and Continuous Graphs

• Polynomial functions of degree $2$ or higher (quadratics, cubics, etc.) have graphs that are intrinsically smooth and continuous.

• Smooth: This characteristic means the graphs contain only rounded curves, transitioning fluidly without any sudden changes in direction. There are absolutely no sharp corners or cusps, unlike functions such as absolute value functions (e.g., $f(x) = |x|$) which have a sharp V-shape.

• Continuous: This characteristic means the graphs possess no breaks, holes, or jumps. You can trace the entire graph of a polynomial function without lifting your pencil from the rectangular coordinate system. This is in contrast to rational functions, which often have vertical asymptotes (breaks) or holes, or step functions which have jumps.

III. End Behavior of Polynomial Functions

• End behavior refers specifically to the behavior of the graph of a function as $x$ approaches positive infinity (moving far to the right on the x-axis) or negative infinity (moving far to the left on the x-axis). It describes whether the graph ultimately rises or falls on either side.

• The graph of any polynomial function will always either rise or fall at each end. It never levels off or oscillates indefinitely.

• The Leading Coefficient Test: This test is a fundamental tool for quickly determining the end behavior of the graph of a polynomial function $f(x) = a<em>n x^n + a</em>{n-1} x^{n-1} + \cdots + a<em>2 x^2 + a</em>1 x + a<em>0$ as $x$ increases or decreases without bound. The behavior is solely dictated by the degree (n) and the sign of the leading coefficient ($a</em>n$).

  1. If $n$ (degree) is odd: (Similar to $y=x$ or $y=x^3$)

     • If the leading coefficient a_n > 0 (positive), the graph resembles a positive odd power function: it falls to the left (as $x \to -\infty, f(x) \to -\infty$) and rises to the right (as $x \to \infty, f(x) \to \infty$).

     • If the leading coefficient a_n < 0 (negative), the graph resembles a negative odd power function: it rises to the left (as $x \to -\infty, f(x) \to \infty$) and falls to the right (as $x \to \infty, f(x) \to -\infty$).

  2. If $n$ (degree) is even: (Similar to $y=x^2$ or $y=x^4$)

     • If the leading coefficient a_n > 0 (positive), the graph resembles a positive even power function: it rises to the left (as $x \to -\infty, f(x) \to \infty$) and rises to the right (as $x \to \infty, f(x) \to \infty$).

     • If the leading coefficient a_n < 0 (negative), the graph resembles a negative even power function: it falls to the left (as $x \to -\infty, f(x) \to -\infty$) and falls to the right (as $x \to \infty, f(x) \to -\infty$).

• Example 3: Use the Leading Coefficient Test for $f(x) = x^3 + 3x^2 - x - 3$.

  • The degree $n=3$ (odd).

  • The leading coefficient $a_n=1$ (positive).

  • Therefore, based on the test for odd degrees with a positive leading coefficient, the graph falls to the left and rises to the right.

• Example 4: Use the Leading Coefficient Test for $f(x) = -4x^3(x - 1)^2(x + 5)$.

  • First, we need to determine the overall degree by identifying the highest power of $x$ when the polynomial is fully expanded. This is done by adding the exponents of the $x$ terms from each factor: $x^3 \cdot (x)^2 \cdot x^1 = x^{3+2+1} = x^6$. So, the degree $n=6$ (even).

  • Next, determine the leading coefficient by multiplying the leading coefficients of each factor: The leading coefficient is $-4$ (from $-4x^3$) multiplied by $(1)^2$ (from $(x-1)^2$) multiplied by $1$ (from $(x+5)$), which gives $-4 \cdot 1 \cdot 1 = -4$. So, the leading coefficient $a_n = -4$ (negative).

  • Therefore, according to the test for even degrees with a negative leading coefficient, the graph falls to the left and falls to the right.

IV. Zeros of Polynomial Functions and Multiplicity

• Zeros of $f$: If $f$ is a polynomial function, the values of $x$ for which $f(x) = 0$ are known as the zeros of $f$. These values are also referred to as the roots or solutions of the polynomial equation $f(x) = 0$. Graphically, these zeros correspond to the x-intercepts where the graph crosses or touches the x-axis. A polynomial of degree $n$ can have at most $n$ real zeros.

• Example 5: Find all zeros of $f(x) = x^3 + 3x^2 - x - 3$.

  • Factor the polynomial by grouping:
    $f(x) = x^2(x+3) - 1(x+3)$
    $f(x) = (x^2-1)(x+3)$
    $f(x) = (x-1)(x+1)(x+3)$

  • Set each factor to zero to find the zeros:
    $x-1=0 \Rightarrow x=1$
    $x+1=0 \Rightarrow x=-1$
    $x+3=0 \Rightarrow x=-3$

  • The zeros are $1, -1, -3$. These are the x-intercepts of the graph where the function's value is zero.

• Multiplicities of Zeros: When factoring the equation for a polynomial function $f$, if the same factor $(x-r)$ occurs $k$ times, but not $k+1$ times, we say that $r$ is a zero with multiplicity $k$. The multiplicity of a zero significantly impacts how the graph behaves at the x-axis.

  • Even Multiplicity: If $r$ is a zero of even multiplicity (e.g., $k=2, 4, 6, \dots$), then the graph touches the x-axis and turns around at $r$. The function's sign (positive or negative) does not change at this zero; it maintains the same sign on both sides of $r$. This is similar to the behavior of a parabola at its vertex on the x-axis (e.g., $y=(x-r)^2$).

  • Odd Multiplicity: If $r$ is a zero of odd multiplicity (e.g., $k=1, 3, 5, \dots$), then the graph crosses the x-axis at $r$. The function's sign changes at this zero (from positive to negative, or vice-versa), similar to how a cubic function behaves at an x-intercept (e.g., $y=(x-r)^3$).

• Example 6: Find the zeros of $f(x) = \frac{1}{2}(x+1)(2x - 3)^2$ and give the multiplicity of each zero. State whether the graph crosses the x-axis or touches the x-axis and turns around at each zero.

  • For the factor $(x+1)$: Set $x+1=0 \Rightarrow x=-1$. This zero has a multiplicity of $1$ (odd, as the exponent implied is $1$). Therefore, the graph crosses the x-axis at $x=-1$.

  • For the factor $(2x-3)^2$: Set $2x-3=0 \Rightarrow x=\frac{3}{2}$. This zero has a multiplicity of $2$ (even). Therefore, the graph touches the x-axis and turns around at $x=\frac{3}{2}$.

• Example 7: Find the zeros of $f(x) = x^3 - 8x^2 + 16x$ and give the multiplicity of each zero. State whether the graph crosses the x-axis or touches the x-axis and turns around at each zero.

  • Factor out the common factor $x$:
    $f(x) = x(x^2 - 8x + 16)$

  • Factor the quadratic expression (which is a perfect square trinomial):
    $f(x) = x(x-4)^2$

  • For the factor $x$: Set $x=0$. This zero has a multiplicity of $1$ (odd). Therefore, the graph crosses the x-axis at $x=0$.

  • For the factor $(x-4)^2$: Set $x-4=0 \Rightarrow x=4$. This zero has a multiplicity of $2$ (even). Therefore, the graph touches the x-axis and turns around at $x=4$.

• Example 8: Write an equation of a polynomial function with the given characteristics: The graph touches the x-axis and turns at $0$ and crosses the x-axis at $6$; the graph lies below the x-axis between $0$ and $6$.

  • Touches and turns at $x=0$: This indicates that $(x-0)^k = x^k$ is a factor, and $k$ must be an even multiplicity. The simplest case is $k=2$. So, a factor is $x^2$.

  • Crosses the x-axis at $x=6$: This indicates that $(x-6)^m$ is a factor, and $m$ must be an odd multiplicity. The simplest case is $m=1$. So, a factor is $(x-6)$.

  • Combining these, a preliminary form of the function is $f(x) = a \cdot x^2 (x-6)$, where $a$ is the leading coefficient that determines the overall scaling and vertical orientation of the graph.

  • Lies below the x-axis between $0$ and $6$: To determine the sign of $a$, we test a point within the interval $(0, 6)$, for instance, $x=1$:
    $f(1) = a \cdot (1)^2 (1-6) = a \cdot 1 \cdot (-5) = -5a$.

  • For the function to lie below the x-axis (f(x) < 0) between $0$ and $6$, we need f(1) < 0. So, -5a < 0. Dividing by $-5$ and reversing the inequality sign gives a > 0. The simplest positive value for $a$ is $1$.

  • Therefore, addressing all conditions, a possible equation for the polynomial function is $f(x) = 1 \cdot x^2(x-6) = x^2(x-6)$, which can also be expanded to $f(x) = x^3 - 6x^2$. Any positive constant multiple of this function would also satisfy the given conditions (e.g., $f(x) = 2x^2(x-6)$).

V. The Intermediate Value Theorem

• Theorem 9 (The Intermediate Value Theorem for Polynomials): Let $f$ be a polynomial function with real coefficients. If $f(a)$ and $f(b)$ have opposite signs (meaning one function value is positive and the other is negative) for real numbers $a$ and $b$ with a < b, then there exists at least one real value $c$ between $a$ and $b$ for which $f(c) = 0$. Equivalently, the polynomial equation $f(x) = 0$ has at least one real root (or zero) situated between $a$ and $b$. This theorem relies on the continuous nature of polynomial graphs.

• This theorem is exceptionally useful for locating real zeros of polynomial functions within specific intervals. It doesn't tell us what the zero is, only that one exists when a sign change occurs in the function's value over an interval, acting as an existence theorem.

• Example 10: Show that the polynomial function $f(x) = x^3 - 2x - 5$ has a real zero between $2$ and $3$.

  • First, evaluate the function at the lower bound of the interval, $x=2$:
    $f(2) = (2)^3 - 2(2) - 5 = 8 - 4 - 5 = -1$

  • Next, evaluate the function at the upper bound of the interval, $x=3$:
    $f(3) = (3)^3 - 2(3) - 5 = 27 - 6 - 5 = 16$

  • Since $f(2) = -1$ (a negative value) and $f(3) = 16$ (a positive value), $f(a)$ and $f(b)$ have opposite signs. Because polynomial functions are continuous, for the graph to go from below the x-axis (at $x=2$) to above the x-axis (at $x=3$), it must cross the x-axis at least once within that interval.