Prepare for your exams
Get points
Guidelines and tips

Prepare for your exams

Study with the several resources on Docsity

Earn points to download

Earn points by helping other students or get them with a premium plan

Guidelines and tips

Sell on Docsity

Log in Sign up

Prepare for your exams

Study with the several resources on Docsity

Find documents

Prepare for your exams with the study notes shared by other students like you on Docsity

Search Store documents

The best documents sold by students who completed their studies

Search through all study resources

Docsity AINEW

Summarize your documents, ask them questions, convert them into quizzes and concept maps

Explore questions

Clear up your doubts by reading the answers to questions asked by your fellow students

Earn points to download

Earn points by helping other students or get them with a premium plan

Share documents

20 Points

For each uploaded document

Answer questions

5 Points

For each given answer (max 1 per day)

All the ways to get free points

Get points immediately

Choose a premium plan with all the points you need

Study Opportunities

Choose your next study program

Get in touch with the best universities in the world. Search through thousands of universities and official partners

Community

Ask the community

Ask the community for help and clear up your study doubts

University Rankings

Discover the best universities in your country according to Docsity users

Free resources

Our save-the-student-ebooks!

Download our free guides on studying techniques, anxiety management strategies, and thesis advice from Docsity tutors

From our blog

Exams and Study

Go to the blog

Prof. Fowler's Lecture Notes: Polynomial Terminology and Operations, Slides of Algebra

Coba Academy of Beauty and Barbering Algebra

An overview of polynomial terminology, including definitions of coefficients, terms, polynomials, and special terminology such as monomials, binomials, trinomials, degree, leading term, and constant term. The document also covers collecting like terms and the foil method for multiplying binomials.

Typology: Slides

2021/2022

Uploaded on 09/12/2022

electraxx 🇺🇸

4.3

(12)

239 documents

1 / 2

This page cannot be seen from the preview

Don't miss anything!

Prof. Fowler

MAT105: College Algebra

Polynomial Terminology

Coefficient: Number directly in front of a variable by which that variable is multiplied

NOTE: no coefficient implies a coefficient of 1

Term: Product of a coefficient and any number of variables raised to any powers

The parts of a polynomial separated by + or −

Examples: 8

−5𝑥3

3𝑥2𝑦

−4𝜋𝑎𝑏2𝑐3𝑑

Note: Variables should be listed alphabetically, and constants, such as 𝜋, should precede variables

Polynomial: Means “many terms” but can consist of one or more terms

All exponents on all variables are non-negative integers

Examples: 𝑥 − 4

7𝑥4− 5𝑥3−10𝑥2+ 6𝑥 − 8

𝑥3+ 3𝑥2𝑦 + 3𝑥𝑦2+ 𝑦3

3

5𝑛4+ 𝜋𝑥3− 𝑥 + 1.23

√17

Note: Most polynomials in this course will be polynomials in one variable of the following form:

𝑎𝑛𝑥𝑛+ 𝑎𝑛−1𝑥𝑛−1 + 𝑎𝑛−2𝑥𝑛−2 + 𝑎𝑛−3𝑥𝑛−3 + ⋯ + 𝑎3𝑥3+ 𝑎2𝑥2+ 𝑎1𝑥 + 𝑎0

Special Terminology: A polynomial with 1 term is a Monomial

A polynomial with 2 terms is a Binomial

A polynomial with 3 terms is a Trinomial

Degree (Order) of a Term: Sum of all exponents on all variables in the term

NOTE: no exponent implies an exponent of 1

Examples: 8 has degree 0 (no variables)

−5𝑥3 has degree 3

3𝑥2𝑦 has degree 3

−4𝜋𝑎𝑏2𝑐3𝑑 has degree 7

Special Terminology: A term with degree 0 is a Constant Term (or simply Constant)

A term with degree 1 is a Linear Term

A term with degree 2 is a Quadratic Term

A term with degree 3 is a Cubic Term

A term with degree 4 is a Quartic Term

A term with degree 5 is a Quintic Term

Partial preview of the text

Download Prof. Fowler's Lecture Notes: Polynomial Terminology and Operations and more Slides Algebra in PDF only on Docsity!

Prof. Fowler MAT105: College Algebra Polynomial Terminology Coefficient: Number directly in front of a variable by which that variable is multiplied NOTE: no coefficient implies a coefficient of 1 Term: Product of a coefficient and any number of variables raised to any powers The parts of a polynomial separated by + or − Examples: 8 −5𝑥^3 3𝑥^2 𝑦 −4𝜋𝑎𝑏^2 𝑐^3 𝑑 Note: Variables should be listed alphabetically, and constants, such as 𝜋, should precede variables Polynomial: Means “many terms” but can consist of one or more terms All exponents on all variables are non-negative integers Examples: 𝑥 − 4 7𝑥^4 − 5𝑥^3 − 10𝑥^2 + 6𝑥 − 8 𝑥^3 + 3𝑥^2 𝑦 + 3𝑥𝑦^2 + 𝑦^3 3 5

𝑛^4 + 𝜋𝑥^3 − 𝑥 + 1.

Note: Most polynomials in this course will be polynomials in one variable of the following form: 𝑎𝑛𝑥𝑛^ + 𝑎𝑛−1𝑥𝑛−1^ + 𝑎𝑛−2𝑥𝑛−2^ + 𝑎𝑛−3𝑥𝑛−3^ + ⋯ + 𝑎 3 𝑥^3 + 𝑎 2 𝑥^2 + 𝑎 1 𝑥 + 𝑎 0 Special Terminology: A polynomial with 1 term is a Monomial A polynomial with 2 terms is a Binomial A polynomial with 3 terms is a Trinomial Degree (Order) of a Term: Sum of all exponents on all variables in the term NOTE: no exponent implies an exponent of 1 Examples: 8 has degree 0 (no variables) −5𝑥^3 has degree 3 3𝑥^2 𝑦 has degree 3 −4𝜋𝑎𝑏^2 𝑐^3 𝑑 has degree 7 Special Terminology: A term with degree 0 is a Constant Term (or simply Constant ) A term with degree 1 is a Linear Term A term with degree 2 is a Quadratic Term A term with degree 3 is a Cubic Term A term with degree 4 is a Quartic Term A term with degree 5 is a Quintic Term

Prof. Fowler Leading Term: Term with highest degree NOTE: does not have to be written first (but should be in most cases) Leading Coefficient: Coefficient of leading term Degree (Order) of a Polynomial: Degree of leading term Note: Polynomials are classified by degrees using same terminology as for terms if possible Examples: 5𝑥^2 − 6𝑥 + 7 is a Quadratic Polynomial 10𝑥^5 + 3 is a Quintic Polynomial 4𝑥^8 − 3𝑥^5 + 2𝑥^4 − 𝑥 − 9 is an 8th-Degree Polynomial Descending Order: Writing a polynomial such that the degrees of its terms decrease from left to right Note: Standard convention is to write polynomials in descending order Example: 6𝑥 + 5𝑥^3 − 2 − 9𝑥^2 should be written as 5𝑥^3 − 9𝑥^2 + 6𝑥 − 2 Ascending Order: Writing a polynomial such that the degrees of its terms increase from left to right Note: Ascending order is often used if leading term is negative and lowest degree term is positive Example: −2𝑥^3 − 𝑥^2 + 5𝑥 + 7 can be written as 7 + 5𝑥 − 𝑥^2 − 2𝑥^3 Like (Similar) Terms: Terms with same exact variables with same exact corresponding exponents Examples of like terms: 3𝑥^2 and −5𝑥^2 14𝑥𝑦^2 𝑧^4 and 8𝑦^2 𝑥𝑧^4 Examples of non-like terms: 4𝑥^3 and 4𝑦^3 𝑎𝑏^2 𝑐^3 and 𝑎^3 𝑏𝑐^2 Collecting (Combining) Like Terms: Adding coefficients of like terms to obtain one simplified result Examples: 9𝑥^2 + 5𝑥^3 + 6𝑥^3 + 3𝑥^2 = 12𝑥^2 + 11𝑥^3 8𝑎^2 𝑏 − 7𝑎𝑏^2 − 4𝑎𝑏 + 3𝑎^2 𝑏 − 5𝑎𝑏 = 11𝑎^2 𝑏 − 7𝑎𝑏^2 − 9𝑎𝑏 FOIL: Acronym for order in which terms of two binomials are multiplied to obtain product F: First (multiply first terms in respective sets of parentheses) O: Outer/Outside (multiply first term in first set of parentheses by second term in second set) I: Inner/Inside (multiply second term in first set of parentheses by first term in second set) L: Last (multiply second terms in respective sets of parentheses) Note: As with all work with polynomials, combine like terms in final result if possible Examples: (𝑥 + 5)(𝑥 − 3) = 𝑥^2 − 3𝑥 + 5𝑥 − 15 = 𝑥^2 + 2𝑥 − 15 (2𝑎𝑏 − 7𝑐)(6𝑏^3 + 1) = 12𝑎𝑏^4 + 2𝑎𝑏 − 42𝑏^3 𝑐 − 7𝑐 = 12𝑎𝑏^4 − 42𝑏^3 𝑐 + 2𝑎𝑏 − 7𝑐

Prof. Fowler's Lecture Notes: Polynomial Terminology and Operations, Slides of Algebra

Related documents

Partial preview of the text

Download Prof. Fowler's Lecture Notes: Polynomial Terminology and Operations and more Slides Algebra in PDF only on Docsity!

𝑛^4 + 𝜋𝑥^3 − 𝑥 + 1.