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

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2021/2022

Uploaded on 09/12/2022

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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 โˆ’
Examples: 8
โˆ’5๐‘ฅ3
3๐‘ฅ2๐‘ฆ
โˆ’4๐œ‹๐‘Ž๐‘2๐‘3๐‘‘
Note: Variables should be listed alphabetically, and constants, such as ๐œ‹, should precede variables
Polynomial: Means โ€œmany termsโ€ but can consist of one or more terms
All exponents on all variables are non-negative integers
Examples: ๐‘ฅ โˆ’ 4
7๐‘ฅ4โˆ’ 5๐‘ฅ3โˆ’10๐‘ฅ2+ 6๐‘ฅ โˆ’ 8
๐‘ฅ3+ 3๐‘ฅ2๐‘ฆ + 3๐‘ฅ๐‘ฆ2+ ๐‘ฆ3
3
5๐‘›4+ ๐œ‹๐‘ฅ3โˆ’ ๐‘ฅ + 1.23
โˆš17
Note: Most polynomials in this course will be polynomials in one variable of the following form:
๐‘Ž๐‘›๐‘ฅ๐‘›+ ๐‘Ž๐‘›โˆ’1๐‘ฅ๐‘›โˆ’1 + ๐‘Ž๐‘›โˆ’2๐‘ฅ๐‘›โˆ’2 + ๐‘Ž๐‘›โˆ’3๐‘ฅ๐‘›โˆ’3 + โ‹ฏ + ๐‘Ž3๐‘ฅ3+ ๐‘Ž2๐‘ฅ2+ ๐‘Ž1๐‘ฅ + ๐‘Ž0
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)
โˆ’5๐‘ฅ3 has degree 3
3๐‘ฅ2๐‘ฆ has degree 3
โˆ’4๐œ‹๐‘Ž๐‘2๐‘3๐‘‘ 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
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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 โˆ’ Examples: 8 โˆ’5๐‘ฅ^3 3๐‘ฅ^2 ๐‘ฆ โˆ’4๐œ‹๐‘Ž๐‘^2 ๐‘^3 ๐‘‘ Note: Variables should be listed alphabetically, and constants, such as ๐œ‹, should precede variables Polynomial: Means โ€œmany termsโ€ but can consist of one or more terms All exponents on all variables are non-negative integers Examples: ๐‘ฅ โˆ’ 4 7๐‘ฅ^4 โˆ’ 5๐‘ฅ^3 โˆ’ 10๐‘ฅ^2 + 6๐‘ฅ โˆ’ 8 ๐‘ฅ^3 + 3๐‘ฅ^2 ๐‘ฆ + 3๐‘ฅ๐‘ฆ^2 + ๐‘ฆ^3 3 5

๐‘›^4 + ๐œ‹๐‘ฅ^3 โˆ’ ๐‘ฅ + 1.

Note: Most polynomials in this course will be polynomials in one variable of the following form: ๐‘Ž๐‘›๐‘ฅ๐‘›^ + ๐‘Ž๐‘›โˆ’1๐‘ฅ๐‘›โˆ’1^ + ๐‘Ž๐‘›โˆ’2๐‘ฅ๐‘›โˆ’2^ + ๐‘Ž๐‘›โˆ’3๐‘ฅ๐‘›โˆ’3^ + โ‹ฏ + ๐‘Ž 3 ๐‘ฅ^3 + ๐‘Ž 2 ๐‘ฅ^2 + ๐‘Ž 1 ๐‘ฅ + ๐‘Ž 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) โˆ’5๐‘ฅ^3 has degree 3 3๐‘ฅ^2 ๐‘ฆ has degree 3 โˆ’4๐œ‹๐‘Ž๐‘^2 ๐‘^3 ๐‘‘ 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: 5๐‘ฅ^2 โˆ’ 6๐‘ฅ + 7 is a Quadratic Polynomial 10๐‘ฅ^5 + 3 is a Quintic Polynomial 4๐‘ฅ^8 โˆ’ 3๐‘ฅ^5 + 2๐‘ฅ^4 โˆ’ ๐‘ฅ โˆ’ 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: 6๐‘ฅ + 5๐‘ฅ^3 โˆ’ 2 โˆ’ 9๐‘ฅ^2 should be written as 5๐‘ฅ^3 โˆ’ 9๐‘ฅ^2 + 6๐‘ฅ โˆ’ 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: โˆ’2๐‘ฅ^3 โˆ’ ๐‘ฅ^2 + 5๐‘ฅ + 7 can be written as 7 + 5๐‘ฅ โˆ’ ๐‘ฅ^2 โˆ’ 2๐‘ฅ^3 Like (Similar) Terms: Terms with same exact variables with same exact corresponding exponents Examples of like terms: 3๐‘ฅ^2 and โˆ’5๐‘ฅ^2 14๐‘ฅ๐‘ฆ^2 ๐‘ง^4 and 8๐‘ฆ^2 ๐‘ฅ๐‘ง^4 Examples of non-like terms: 4๐‘ฅ^3 and 4๐‘ฆ^3 ๐‘Ž๐‘^2 ๐‘^3 and ๐‘Ž^3 ๐‘๐‘^2 Collecting (Combining) Like Terms: Adding coefficients of like terms to obtain one simplified result Examples: 9๐‘ฅ^2 + 5๐‘ฅ^3 + 6๐‘ฅ^3 + 3๐‘ฅ^2 = 12๐‘ฅ^2 + 11๐‘ฅ^3 8๐‘Ž^2 ๐‘ โˆ’ 7๐‘Ž๐‘^2 โˆ’ 4๐‘Ž๐‘ + 3๐‘Ž^2 ๐‘ โˆ’ 5๐‘Ž๐‘ = 11๐‘Ž^2 ๐‘ โˆ’ 7๐‘Ž๐‘^2 โˆ’ 9๐‘Ž๐‘ 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: (๐‘ฅ + 5)(๐‘ฅ โˆ’ 3) = ๐‘ฅ^2 โˆ’ 3๐‘ฅ + 5๐‘ฅ โˆ’ 15 = ๐‘ฅ^2 + 2๐‘ฅ โˆ’ 15 (2๐‘Ž๐‘ โˆ’ 7๐‘)(6๐‘^3 + 1) = 12๐‘Ž๐‘^4 + 2๐‘Ž๐‘ โˆ’ 42๐‘^3 ๐‘ โˆ’ 7๐‘ = 12๐‘Ž๐‘^4 โˆ’ 42๐‘^3 ๐‘ + 2๐‘Ž๐‘ โˆ’ 7๐‘