1.02 Polynomial Functions Notes

1) Overview and Lesson Goals

This lesson centers on polynomial functions. By the end, you should be able to write a polynomial function, match it to its graph, identify the key points on the graph, and use polynomials to model real-life situations. The class activities include vocabulary review, exploring the factor theorem and the fundamental theorem of algebra, graphing polynomials, and applying polynomial concepts to practical problems. 2) Vocabulary and Core Concepts

Polynomials are expressions that can be written as a sum of a finite number of terms. Each term has a coefficient and a variable raised to a nonnegative integer power. In standard form, the exponents decrease from left to right, but terms with zero coefficients may be omitted. A general polynomial in one variable can be written as

$f(x)=a<em>n x^n + a</em>{n-1} x^{n-1} + \cdots + a<em>1 x + a</em>0,$

where $n$ is a nonnegative integer, $a<em>i$ are real number coefficients for all $i$, and $a</em>n \neq 0$ (making $a_n$ the leading coefficient and $n$ the degree). In standard form, the exponents appear in descending order, but not all powers must be present; missing terms simply have coefficient zero. Key vocabulary includes:

• Constant: a polynomial with no variable; a single number, e.g. $f(x)=c$.

• Monomial: a polynomial with one term.

• Binomial: a polynomial with two terms.

• Trinomial: a polynomial with three terms.

• Degree: the highest exponent of the polynomial, i.e., the largest $n$ such that $a_n\neq 0$. The degree determines many graph features.

• X-intercept: a point where the graph crosses the x-axis; at an x-intercept the y-coordinate is $0$, so the point is of the form $(-7,0)$ in an example.

• Y-intercept: a point where the graph crosses the y-axis; for a function this occurs at $x=0$, yielding a point of the form $(0,b)$, such as $(0,4)$ in the example.

• Linear function: a polynomial of degree 1; its graph is a line.

• Quadratic function: a polynomial of degree 2; its graph is a parabola.

• Cubic function: a polynomial of degree 3; graph features include at least one turning point; the transcript notes a peak and a valley, though standard theory describes up to two turning points for a cubic.

• Relative minimum: a point where the graph changes from decreasing to increasing, i.e., a valley.

• Relative maximum: a point where the graph changes from increasing to decreasing, i.e., a peak. 3) What is a Polynomial? Examples and Non-Examples

  A polynomial has positive integer exponents and a finite number of terms. It is a sum of terms with nonzero integer exponents. For instance, the following are polynomials:

• $3x - 2$

• $8x^3 - \sqrt{5}$ (here $\sqrt{5}$ is a constant term, so the expression is still a polynomial)

• $2x^2 - 4x - 7$

• $y = 7x^6 + 9x^5 - x^4 + 3x^3$
  Conversely, expressions that involve negative exponents, fractions with the variable in the denominator, or radicals that involve the variable inside the radical are not polynomials. For example, a term with a denominator like $\frac{1}{x}$ or a radical such as $\sqrt{x}$ (where $x$ is inside the radical) prevents the expression from being a polynomial. In standard form, polynomials may omit certain powers entirely (e.g., $x^4 + x^2 + c$ is still a polynomial), because missing powers simply have coefficient $0$. 4) Degree, End Behavior, and Graph Features

  The degree is the highest exponent present in the polynomial, and the leading coefficient is the coefficient of the term with the highest degree ($a_n$ from the general form). These two features crucially guide end behavior, describing the direction of the graph as $x$ approaches positive or negative infinity:

• Even Degree, Positive Leading Coefficient (a_n > 0): Both ends of the graph go up (as $x \to \pm\infty$, $f(x) \to +\infty$).

• Even Degree, Negative Leading Coefficient (a_n < 0): Both ends of the graph go down (as $x \to \pm\infty$, $f(x) \to -\infty$).

• Odd Degree, Positive Leading Coefficient (a_n > 0): The left end goes down, and the right end goes up (as $x \to -\infty$, $f(x) \to -\infty$; as $x \to +\infty$, $f(x) \to +\infty$).

• Odd Degree, Negative Leading Coefficient (a_n < 0): The left end goes up, and the right end goes down (as $x \to -\infty$, $f(x) \to +\infty$; as $x \to +\infty$, $f(x) \to -\infty$).

Multiplicity describes how many times a particular zero occurs. If a zero has even multiplicity, the graph touches the x-axis and bounces off (does not cross) at that point. If a zero has odd multiplicity:

• If the multiplicity is 1, the graph crosses the x-axis linearly.

• If the multiplicity is greater than 1 (e.g., 3, 5), the graph crosses the x-axis but flattens out or has an inflection point at the x-intercept before continuing to cross.
  These ideas help interpret graphs and predict intercept behavior. 5) Zeros, Intercepts, and the Factor Theorem

  Zeros, x-intercepts, and roots are all the same concept: the values of $x$ for which $f(x)=0$. The factor theorem provides a practical test: if $f(k)=0$ for some $k$, then $(x-k)$ is a factor of the polynomial $f(x)$. The fundamental theorem of algebra states that a polynomial of degree $n$ has exactly $n$ roots (zeros) in the complex number system, counting multiplicities. In particular, a cubic (degree 3) has three roots in total (real or complex), counting multiplicities. To find zeros, several approaches can be used:

  • Factor theorem: factor $f(x)$, set each factor equal to zero, and solve for $x$.

  • Graphing: locate x-intercepts on a graph or calculator and read off zeros.

  • Synthetic division: divide by a proposed factor to check for a zero (no remainder indicates a factor). The lesson notes that there are often multiple valid approaches; you should use the method you know best unless a specific method is requested. 6) Intercepts and an Example from a Table

  In a classroom activity, a table of values was used to identify the x- and y-intercepts, as well as the relative maximum and minimum values:

  • Y-intercept: This occurs where the graph crosses the y-axis, meaning the x-coordinate is $0$. The coordinate is of the form $(0,b)$. In the example, the y-intercept was $(0,4)$.

  • X-intercepts (Zeros): These occur where the graph crosses or touches the x-axis, meaning the y-coordinate is $0$. The coordinates are of the form $(a,0)$. In the example, an x-intercept was $(-7,0)$ (the only one shown in that table).

  • Relative Maximum and Minimum: These refer to local peaks and valleys in the graph. From a table, a relative minimum is identified when the y-values decrease and then start increasing again (a 'valley'). The minimum value given in the example was $(-4,-12)$. A relative maximum is identified when y-values increase and then start decreasing (a 'peak'); while the explicit coordinate for the maximum was not provided, it would be the point $(x,y)$ where $y$ is momentarily the highest value in its immediate vicinity. 7) Graphing Polynomials: Multiplicity and End Behavior in Practice

  A key activity involved choosing which graph correctly represents a given polynomial, using zeros and their multiplicities as clues. For example, a factor such as $(x+2)^2$ indicates a zero at $x=-2$ with even multiplicity, meaning the graph touches the x-axis at that zero and turns around. A simple zero like $(x-1)$ indicates multiplicity 1, so the graph crosses the axis there. When selecting a graph, you can also use the leading coefficient sign to infer end behavior: if the degree is odd and the leading coefficient is negative, the left end goes up while the right end goes down; if the degree is even with a negative leading coefficient, both ends go down; with positive leading coefficient, both ends go up for even degree, and for odd degree the ends go in opposite directions. The transcript shows practice with a specific polynomial of the form

  $h(x)= -(x+2)^2 (x-1) (x-4)^2,$

  where the multiplicities are evident from the exponents: a double root at $x=-2$, a single root at $x=1$ (odd multiplicity), and a double root at $x=4$ (even multiplicity). The graph is analyzed by locating these zeros and their multiplicities and then checking end behavior from the overall degree and leading coefficient. 8) A Cubic Polynomial with Zeros at 2, $\pm\sqrt{3}$

  Using the factor theorem and standard form, a polynomial with zeros at $x=2$, $x=\sqrt{3}$, and $x=-\sqrt{3}$ can be constructed. Because the roots come in conjugate pairs when coefficients are real, a convenient polynomial (up to a constant factor) is

  $f(x) = (x-2)(x-\sqrt{3})(x+\sqrt{3}) = (x-2)(x^2-3).$

  Expanding yields

  $f(x) = x^3 - 2x^2 - 3x + 6,$

  which is in standard form with descending powers. Any nonzero constant multiple of this polynomial would also have the same zeros, e.g., $k(x^3 - 2x^2 - 3x + 6)$ for $k\neq 0$. 9) Application Problem: Boxes and Volume

  A real-world application problem involves designing a cardboard box with a fixed volume. Given the volume requirement

  $V=35\text{ cubic inches},$

  let the width be $w$ inches. The length is five inches less than three times the width, so

  $L = 3w - 5,$

  and the height is two inches less than the width, so

  $H = w - 2.$

  The volume equation is then

  $V = L\times W\times H = (3w-5)\, w \, (w-2) = 35.$

  Expanding and simplifying gives the cubic equation

  $3w^3 - 11w^2 + 10w - 35 = 0.$

  Solving for $w$ (by factoring if possible or numerical methods) yields a width of approximately

  $w \approx 3.633\text{ inches}.$

  Since width is a measurement with units, the final answer includes units. The transcript notes that a numerical solution is obtained, highlighting the step of translating the word problem into a polynomial equation and then solving it to obtain the width. 10) Desmos and Practice Modes

  Throughout the lesson, Desmos was used for practice, including a card-matching activity to reinforce vocabulary (matching terms to definitions or graphs) and an exercise to identify zeros of a function. The instructor reminded students that Desmos is a tool to practice and that it
  care okay to try different attempts and correct mistakes as part of learning. The class also revisited the idea that there can be multiple valid approaches to a problem (e.g., solving for zeros by factoring, graphing, or synthetic division). 11) Connections, Implications, and Takeaways

  • Polynomials form the backbone of many modeling tasks in algebra and precalculus. The definitions and rules—especially about exponents, end behavior, and intercepts—give you a toolkit to analyze and graph these functions quickly.

  • The Factor Theorem and Fundamental Theorem of Algebra connect algebraic structure (factoring) to geometric interpretation (zeros and x-intercepts) and guarantee a predictable count of zeros based on degree (counting multiplicities in the complex plane).

  • Understanding multiplicity helps anticipate how the graph interacts with the x-axis, which is essential when sketching graphs and solving intercept problems.

  • Real-world word problems translate into polynomial equations by identifying given quantities (e.g., volume, length, width, height) and expressing relationships via algebraic expressions, then solving for the unknowns with appropriate algebraic techniques. 12) Quick Reference: Key Formulas and Concepts

  • General polynomial (one variable):

  $f(x) = a<em>n x^n + a</em>{n-1} x^{n-1} + \cdots + a<em>1 x + a</em>0,$

  where $n \in \mathbb{Z}<em>{\ge 0}$ and $a</em>n \neq 0$.

  • Zeros/intercepts: solve $f(x)=0$ for zeros; y-intercept occurs at $x=0$; x-intercepts occur where $f(x)=0$.

  • Factor Theorem: If $f(k)=0$, then $x-k$ is a factor of $f(x)$.

  • Fundamental Theorem of Algebra: A polynomial of degree $n$ has exactly $n$ roots (counting multiplicities) in the complex numbers.

  • Multiplicity: if a zero occurs with even multiplicity, the graph touches the x-axis; if it occurs with odd multiplicity: the graph crosses the x-axis (crossing linearly if multiplicity is 1, or flattening if multiplicity is greater than 1).

  • End behavior:

  • Even Degree, Positive Leading Coefficient (a_n > 0): As $x \to \pm\infty$, $f(x) \to +\infty$.

  • Even Degree, Negative Leading Coefficient (a_n < 0): As $x \to \pm\infty$, $f(x) \to -\infty$.

  • Odd Degree, Positive Leading Coefficient (a_n > 0): As $x \to -\infty$, $f(x) \to -\infty$ and as $x \to +\infty$, $f(x) \to +\infty$.

  • Odd Degree, Negative Leading Coefficient (a_n < 0): As $x \to -\infty$, $f(x) \to +\infty$ and as $x \to +\infty$, $f(x) \to -\infty$.

  • Example of zeros from factors: zeros at 2, $\pm\sqrt{3}$ yield

  $f(x) = (x-2)(x-\sqrt{3})(x+\sqrt{3}) = (x-2)(x^2-3).$ Note: The transcript contains several worked examples and class activities intended to reinforce these concepts, including identifying polynomials from lists of expressions, using Desmos for practice, solving zeros with the factor theorem, and applying polynomial reasoning to a real-world volume problem. These notes capture the major and minor points presented for 1.02 and provide a consolidated reference for exam preparation.