Lecture Notes: Polynomials and WebAssign Overview

Canvas and WebAssign Setup

• Canvas course published; log in to Canvas and check the home page.
• Announcement posted with steps to register for WebAssign.
• WebAssign usage in course: weekly assignments, tests.
• Registration steps for WebAssign:
  • Go to cengage.com.
  • Register with your VT email as a student account.
  • Return to Canvas and click on an assignment to complete registration for the course.
• Files on Canvas:
  • Cost policy (document with required textbook and assessment details).
  • Lecture note outlines (download before class; notes filled in during lecture).
• Lectures are recorded or notes posted; completed notes posted at end of week.

Course Structure and Grading

• Schedule: three lectures per week on Monday, Wednesday, Friday at 10:10 AM.
• Recitation session: scheduled at a different time than the 10:10 meeting; may be on Mon/Wed/Fri; attendance/time shown in schedule.
• This week: no recitation.
• Recitation assessment: attendance and participation account for 5% of the final grade.
  • You receive a worksheet in recitation; TA checks completion; counts toward the semester grade.
• Assessments:
  • Three in-class tests (all in class, use your own laptop, through WebAssign).
  • One final exam.
  • Each test: 15% of the grade.
  • Final exam: 20% of the grade.
• Homework and WebAssign:
  • Weekly WebAssign assignments: 15% of the grade.
  • Weekly homework (typically due Friday): 10% of the grade.
• Quizzes:
  • Well, assigned quizzes; conducted before test days to provide a brief review.
• Pace and expectations:
  • Course moves at a fast pace; stay up-to-date with material.
• If questions arise, ask in class; the instructor will publish modules after setup.

Exponential Notation and Positive Integers (Section 1.2 groundwork)

• Exponential notation basics:
  • For a real number a and a positive integer n, the expression a^n means multiplying a by itself n times.
  • In this context, n is called the exponent (power) and a is the base.
• Special case: zero exponent:
  • For a real number x ≠ 0, x^0 = 1.
• Examples (non-negative integer exponents):
  • 5^3 = 5 imes 5 imes 5 = 125
  • (-2)^4 = (-2) imes (-2) imes (-2) imes (-2) = 16
  • If the negative is not inside parentheses: ( -2 )^4 = 16 ext{ vs } -2^4 = -(2^4) = -16
  • Fractions: ( frac{1}{4})^4 = frac{1}{256}
• Basic exponent rule (same base):
  • If m and n are positive integers, then
  • a^m imes a^n = a^{m+n}
• Practical use: combine exponents when multiplying same-base powers.
• Examples applying the rule (brief recap):
  • $3^2 imes 3^2 = 3^{4}$.
  • $7^2 imes 7^5 = 7^{7}$.
  • For a more complex base, e.g., $(7 imes 8)^5$ or $(7 imes a)^5$, the same exponent-addition rule applies to the shared base when factored properly.

Polynomials: Definitions and Terminology

• Variable:
  • A letter that stands for a real number.
• Polynomial (one variable):
  • A sum of powers of a variable with nonnegative integer exponents.
  • Form: sum of terms of the form coefficient × x^k where k ∈ {0,1,2,…}.
• Degree of a polynomial:
  • The highest exponent appearing in the polynomial.
  • Convention: write polynomials with descending powers of the variable.
• Constant polynomial:
  • A polynomial with degree 0 (e.g., 19).
  • Reason: 19 can be written as 19 × x^0, so its degree is 0.
• Not polynomials:
  • Examples failing the nonnegative integer exponent condition:
  • 7x^{-1} (negative exponent).
  • x^{1/2} (fractional exponent).
  • Rational expressions (ratios of polynomials) are not polynomials.
• Terms and coefficients:
  • Each addend in a polynomial is a term.
  • The real numbers in front of the powers are the coefficients.
  • Coefficients can be arbitrary real numbers (negative, irrational, fractions, etc.).
• Polynomials in two variables:
  • Expression of the form ext{coeff} imes x^a y^b summed over various terms.
  • Variables do not have to appear in every term, but all variables must appear in at least one term.
  • Degree concept extended: for a term x^a y^b, its degree is a + b.
  • Degree of a multivariable polynomial is the maximum degree across all its terms.
• Examples (two variables): polynomials in x and y or in u and v, etc.
• Degree examples (two variables):
  • Term x^4 y^6 has degree 4+6 = 10.
  • Term 2 x y^3 has degree 1+3 = 4.
  • Term 2 x y has degree 1+1 = 2.
  • Term x^5 has degree 5.
  • Constant term -7 has degree 0.
  • Therefore the degree of the polynomial is the maximum of these term degrees (e.g., 10 in the first example).
• Degree in three variables:
  • For a term x^a y^b z^c, degree is a + b + c.
  • Example degrees: 3 + 1 + 2 = 6; other terms may have degrees 5, 7, 9; the polynomial degree is the max (e.g., 9 in the given example).

How to Add and Subtract Polynomials

• Key idea: combine like terms.
• Like terms:
  • Terms with exactly the same variables raised to the same powers (e.g., x^3 and x^3, or x^2 y and x^2 y).
• Subtlety: subtracting can be viewed as adding the additive inverse: a − b = a + (−b).
• Example 4a (one-variable polynomials):
  • (-6) x^3 + 17 x^3 + 5 x^2 + 2 x^2 − 8 x − 4
  • Like terms combined: 11 x^3 + 7 x^2 − 12 x − 4.
• Example 4b (subtraction):
  • Subtract a second polynomial by distributing the negative sign.
  • Result given in transcript: 12 x^3 + 4 x^2 + 12 x − 14.
• Combining like terms in two-variable polynomials (examples 4d, 4e):
  • Example d (polynomials in x and y): combine terms with the same x and y exponents.
  • Result given: 7 x^2 y − 4 x y + x.
  • Example e (polynomials in x and y, subtraction): after distributing the negative sign and combining like terms, the final result given is
  • x^4 y^2 + 8 x^3 y + y - 6x.
• Practical tips:
  • When subtracting, you can distribute the negative sign to all terms in the second polynomial before combining.
  • Some students may skip intermediate steps; focus on correctly aligning like terms and summing coefficients.

Multiplying Polynomials and the Distributive Property

• Distributive property (over addition):
  • For any a, b, c: a(b + c) = ab + ac
  • This extends to more than two terms inside the parentheses.
• Distributive property is a key tool for multiplying polynomials.
• Example 5 (three-term distribution):
  • Multiply 3 x^5 by (7 x^3 - 2 x^2 + x - 9).
  • Stepwise expanded product:
  • 3 x^5 imes 7 x^3 = 21 x^8
  • 3 x^5 imes (-2 x^2) = -6 x^7
  • 3 x^5 imes x = 3 x^6
  • 3 x^5 imes (-9) = -27 x^5
  • Final simplified result:
  • 21 x^8 - 6 x^7 + 3 x^6 - 27 x^5.
• Example 5b (distribution with two factors):
  • Can distribute either the entire first factor over the second, or distribute term-by-term (FOIL-style).
  • Demonstrated two methods for (x + 8)(x + 1):
  • Method 1 (full-distribution): x(x + 1) + 8(x + 1) = x^2 + x + 8x + 8 = x^2 + 9x + 8.
  • Method 2 (FOIL-style decomposition): distribute x first, then 8, then combine like terms to get the same result: x^2 + 9x + 8.
• Example 5c (two polynomials with a common factor):
  • Multiply two expressions by distributing, then combine like terms.
  • Transcript final result for this example: 2x^3 + 9x^2 - 10x - 7.
  • Note: The transcript shows a final result; step-by-step arithmetic may depend on the exact second polynomial, but the final expression given is as stated.
• Practical tips:
  • You can distribute to each term in a polynomial on the right (or left) side.
  • Alternative approach: FOIL for binomials; general approach: distribute each term of the first polynomial across every term of the second.
  • With practice, you can skip intermediate steps while keeping track of exponents and coefficients.

Self-Check Exercises and Additional Notes

• The course includes self-check exercises for practice.
• Answers to self-check exercises are posted in an appendix for reference.
• The instructor mentions posting completed notes after class for review.
• Students are encouraged to access the notes ahead of class to follow along more easily.

Practical Reminders and Real-World Relevance

• The course emphasizes exact arithmetic with polynomials, a foundational concept for algebra, calculus, and applied math.
• WebAssign integration highlights the importance of practicing problems in an online environment and the alignment of homework with exams.
• The schedule and grading structure reflect a balance between practice (homework/quizzes) and evaluation (tests/final).
• Understanding polynomials supports modeling real-world quantities, simplifying expressions, and solving equations encountered in engineering, physics, economics, and data science.

Quick Reference (Key Formulas)

• Exponent rule (same base): a^m a^n = a^{m+n}
• Zero exponent: x^0 = 1 ext{ for } x
  eq 0
• Example evaluations: 5^3 = 125,
  ewline (-2)^4 = 16,
  ewline -(2^4) = -16,
  ewline ig( frac{1}{4}ig)^4 = frac{1}{256}
• One-variable polynomial degree: highest exponent in the polynomial.
• Two-variable term degree: for term x^a y^b, degree is a + b; polynomial degree is max over all terms.
• Three-variable term degree: for term x^a y^b z^c, degree is a + b + c; polynomial degree is max over all terms.
• Addition of polynomials: combine like terms.
• Subtraction of polynomials: distribute a negative sign, then combine like terms.
• Multiplication (distributive rule):a(b+c) = ab + ac and extend to more terms; for same base, add exponents when multiplying: a^m a^n = a^{m+n}
• Key example results mentioned in lecture:
  • In-class product: 3 x^5(7 x^3 - 2 x^2 + x - 9) = 21 x^8 - 6 x^7 + 3 x^6 - 27 x^5
  • Binomial product: (x+8)(x+1) = x^2 + 9x + 8
  • Polynomial product (example): final result reported as 2x^3 + 9x^2 - 10x - 7 for a particular setup in the walkthrough