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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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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 โ Examples: 8 โ5๐ฅ^3 3๐ฅ^2 ๐ฆ โ4๐๐๐^2 ๐^3 ๐ Note: Variables should be listed alphabetically, and constants, such as ๐, should precede variables Polynomial: Means โmany termsโ but can consist of one or more terms All exponents on all variables are non-negative integers Examples: ๐ฅ โ 4 7๐ฅ^4 โ 5๐ฅ^3 โ 10๐ฅ^2 + 6๐ฅ โ 8 ๐ฅ^3 + 3๐ฅ^2 ๐ฆ + 3๐ฅ๐ฆ^2 + ๐ฆ^3 3 5
Note: Most polynomials in this course will be polynomials in one variable of the following form: ๐๐๐ฅ๐^ + ๐๐โ1๐ฅ๐โ1^ + ๐๐โ2๐ฅ๐โ2^ + ๐๐โ3๐ฅ๐โ3^ + โฏ + ๐ 3 ๐ฅ^3 + ๐ 2 ๐ฅ^2 + ๐ 1 ๐ฅ + ๐ 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) โ5๐ฅ^3 has degree 3 3๐ฅ^2 ๐ฆ has degree 3 โ4๐๐๐^2 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: 5๐ฅ^2 โ 6๐ฅ + 7 is a Quadratic Polynomial 10๐ฅ^5 + 3 is a Quintic Polynomial 4๐ฅ^8 โ 3๐ฅ^5 + 2๐ฅ^4 โ ๐ฅ โ 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: 6๐ฅ + 5๐ฅ^3 โ 2 โ 9๐ฅ^2 should be written as 5๐ฅ^3 โ 9๐ฅ^2 + 6๐ฅ โ 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: โ2๐ฅ^3 โ ๐ฅ^2 + 5๐ฅ + 7 can be written as 7 + 5๐ฅ โ ๐ฅ^2 โ 2๐ฅ^3 Like (Similar) Terms: Terms with same exact variables with same exact corresponding exponents Examples of like terms: 3๐ฅ^2 and โ5๐ฅ^2 14๐ฅ๐ฆ^2 ๐ง^4 and 8๐ฆ^2 ๐ฅ๐ง^4 Examples of non-like terms: 4๐ฅ^3 and 4๐ฆ^3 ๐๐^2 ๐^3 and ๐^3 ๐๐^2 Collecting (Combining) Like Terms: Adding coefficients of like terms to obtain one simplified result Examples: 9๐ฅ^2 + 5๐ฅ^3 + 6๐ฅ^3 + 3๐ฅ^2 = 12๐ฅ^2 + 11๐ฅ^3 8๐^2 ๐ โ 7๐๐^2 โ 4๐๐ + 3๐^2 ๐ โ 5๐๐ = 11๐^2 ๐ โ 7๐๐^2 โ 9๐๐ 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: (๐ฅ + 5)(๐ฅ โ 3) = ๐ฅ^2 โ 3๐ฅ + 5๐ฅ โ 15 = ๐ฅ^2 + 2๐ฅ โ 15 (2๐๐ โ 7๐)(6๐^3 + 1) = 12๐๐^4 + 2๐๐ โ 42๐^3 ๐ โ 7๐ = 12๐๐^4 โ 42๐^3 ๐ + 2๐๐ โ 7๐