Section 4.1. Polynomial Addition and Subtraction

Section 4.1: Polynomial Addition and subtraction Chapter 4: Polynomial and Radical Operations A monomial is: • • • a constant a variable (where all exponents are whole numbers) product of a constant and a variable. Examples of terms that are monomials: 5, 𝑥 3 , 7𝑥𝑦 2 , 9𝑎, Examples of terms that are not monomials: 5𝑥 −2 , • • • 6 , 𝑥 2 𝑥, 3 −5𝑧 3√𝑥 Variables cannot be in the denominator of a fraction Exponents cannot be negative Variables cannot appear under radicals The coefficient of a monomial with a variable is the number that precedes the variable. The coefficient of a constant monomial is just the constant. Find the coefficient of the following monomials • • • Solution: • • • 3𝑥𝑦 2 −5𝑧 3 −4 3𝑥𝑦 2 coefficient 3 −5𝑧 3 coefficient −5 −4 coefficient −4 The degree of a monomial is the sum of the exponents. The degree of a constant is 0. Find the degree of the following monomials • • • Solution: • 2 3𝑥𝑦 −5𝑧 3 −4 • • 3𝑥𝑦 2 (the x has an implied exponent of 1.) degree = 1+2 = 3 −5𝑧 3 degree = 3 −4 this is a constant, it has degree = 0 Section 4.1: Polynomial Addition and subtraction Chapter 4: Polynomial and Radical Operations Like terms are terms that have the same variables structure. That is, they have the same variables and corresponding variables have the same exponents. The coefficients do not need to match. Not like terms have different variables or corresponding variables have different exponents. Are the terms like terms or not like terms • • • • • • 3xy, 5xy 8𝑎2 , 5𝑎 −4𝑥 2 𝑦, 3𝑥𝑦 7𝑥, 8𝑥 5𝑥 2 , −14𝑥 2 6𝑥, 6𝑦 Solution • • • • • • 3xy, 5xy like terms 8𝑎2 , 5𝑎 not like terms (different exponents) −4𝑥 2 𝑦, 3𝑥𝑦 not like terms (x’s have different exponents) 7𝑥, 8𝑥 like terms 5𝑥 2 , −14𝑥 2 like terms 6𝑥, 6𝑦 not like terms (different variables) To add two or more monomials that are like terms, add the coefficients; keep the variables and exponents on the variables the same. To subtract two or more monomials that are like terms, subtract the coefficients; keep the variables and exponents on the variables the same Add or subtract the monomials • • • • • • • 3x + 5x 8𝑥𝑦 2 + 11𝑥𝑦 2 24𝑧 2 − 15𝑧 2 3𝑥 + 𝑥 + 5𝑥 10𝑦 − 4𝑦 + 3𝑦 8𝑥 2 + 5𝑥 3𝑦 3 − 8𝑦 Solution: • 3x + 5x = 8x • 8𝑥𝑦 2 + 11𝑥𝑦 2 = 19𝑥𝑦 2 • 24𝑧 2 − 15𝑧 2 = 9𝑧 2 • 3𝑥 + 𝑥 + 5𝑥 = 3𝑥 + 1𝑥 + 5𝑥 = 9𝑥 • 10𝑦 − 4𝑦 + 3𝑦 = 9𝑦 • 8𝑥 2 + 5𝑥 can’t be added, not like terms • 3𝑦 3 − 8𝑦 can’t be subtracted, not like terms Section 4.1: Polynomial Addition and subtraction Chapter 4: Polynomial and Radical Operations Polynomial definition A polynomial is a monomial or the sum or difference of monomials. Each monomial is called a term of the polynomial. Important: Terms are separated by addition signs and subtraction signs, but never by multiplication signs A monomial is a polynomial with one term. Here are examples of three different monomials. 8𝑥, 9𝑥𝑦 2 𝑎𝑛𝑑 7 A binomial is the sum or difference of two monomials. Here are examples of three different binomials. 5𝑥 + 7, 𝑥 2 − 4𝑥, 5𝑦 + 9 A trinomial is the sum or difference of three monomials. Here are example of three different monomials. 5𝑥 2 − 6𝑥 + 5, 8𝑥𝑦 2 + 9𝑥𝑦 − 3𝑥, 𝑧 2 − 8𝑧 + 7 • The leading term of a polynomial is the term with the highest degree. • • The leading coefficient of a polynomial is the coefficient of the term with the highest degree. The degree of a polynomial is the degree of the leading term. Find: Solution: to find the leading term Leading term we must compute the degree of Leading coefficient each term. Degree of polynomial • • 5𝑥𝑦 2 + 7𝑥𝑦 − 9𝑦 4 First compute degree of each 2 8𝑥 − 3𝑥 + 7 term 8x is a first-degree term, 5𝑥𝑦 2 third-degree • 8𝑥 − 3𝑥 2 + 7 −3𝑥 2 is a second-degree term 7𝑥𝑦 second-degree • 5𝑥𝑦 2 + 7𝑥𝑦 − 9𝑦 4 7 is a constant and it is a 0-degree −9𝑦 4 fourth degree term Leading term −9𝑦 4 The leading term is −3𝑥 2 (since it Leading coefficient −9 has the highest degree) Degree of polynomial 4 Leading coefficient −3 Degree of polynomial 2 Section 4.1: Polynomial Addition and subtraction Chapter 4: Polynomial and Radical Operations A polynomial is written in standard form provided that the terms are ordered from highest degree to lowest degree. Write the polynomial in standard form: • • Solution • • 5𝑥 − 3 + 8𝑥 3 − 5𝑥 2 7𝑥𝑦 + 8𝑥𝑦 2 − 7𝑥 + 5 8𝑥 3 − 5𝑥 2 + 5𝑥 − 3 8𝑥𝑦 2 + 7𝑥𝑦 − 7𝑥 + 5 To find opposite of a polynomial, you change the sign of each term in the polynomial. Find the opposite of the polynomial • • 8𝑥 2 + 5𝑥 − 3 9𝑥 3 + 7𝑥 2 − 9𝑥 + 3 Solution • • −(8𝑥 2 + 5𝑥 − 3) = −8𝑥 2 − 5𝑥 + 3 −(9𝑥 3 + 7𝑥 2 − 9𝑥 + 3)=−9𝑥 3 − 7𝑥 2 + 9𝑥 − 3 Add the polynomials and write answer in standard form. • • (5𝑥 2 − 7𝑥 + 3) + (8𝑥 2 + 9𝑥 − 10) (2𝑥𝑦 2 + 2𝑥𝑦 + 3𝑥) + (8𝑥𝑦 2 − 4𝑥𝑦 − 10𝑥) Solution: (5𝑥 2 − 7𝑥 + 3) + (8𝑥 2 + 9𝑥 − 10) Parenthesis can be dropped without changing signs. = 5𝑥 2 − 7𝑥 + 3 + 8𝑥 2 + 9𝑥 − 10 = 5𝑥 2 + 8𝑥 2 − 7𝑥 + 9𝑥 + 3 − 10 = 13𝑥 2 + 2𝑥 − 7 (2𝑥𝑦 2 + 2𝑥𝑦 + 3𝑥) + (8𝑥𝑦 2 − 4𝑥𝑦 − 10𝑥) = 2𝑥𝑦 2 + 2𝑥𝑦 + 3𝑥 + 8𝑥𝑦 2 − 4𝑥𝑦 − 10𝑥 = 2𝑥𝑦 2 + 8𝑥𝑦 2 + 2𝑥𝑦 − 4𝑥𝑦 + 3𝑥 − 10𝑥 = 10𝑥𝑦 2 − 2𝑥𝑦 − 7𝑥 Section 4.1: Polynomial Addition and subtraction Subtract the polynomials and write answer in standard form. • • (10𝑥 2 − 2𝑥 − 3) − (3𝑥 2 − 8𝑥 − 10) (2𝑥𝑦 2 + 2𝑥𝑦 + 𝑧 4 ) − (8𝑥𝑦 2 − 4𝑥𝑦 − 2𝑧 4 ) Chapter 4: Polynomial and Radical Operations Solution: (10𝑥 2 − 2𝑥 − 3) − (3𝑥 2 − 8𝑥 − 10) First parenthesis can be dropped with no changes. Need to find opposite when drop second parenthesis. = 10𝑥 2 − 2𝑥 − 3 − 3𝑥 2 + 8𝑥 + 10 = 10𝑥 2 − 3𝑥 2 − 2𝑥 + 8𝑥 − 3 + 10 = 7𝑥 2 + 6𝑥 + 7 (2𝑥𝑦 2 + 2𝑥𝑦 + 𝑧 4 ) − (8𝑥𝑦 2 − 4𝑥𝑦 − 2𝑧 4 ) = 2𝑥𝑦 2 + 2𝑥𝑦 + 1𝑧 4 − 8𝑥𝑦 2 + 4𝑥𝑦 + 2𝑧 4 = 1𝑧 4 + 2𝑧 4 + 2𝑥𝑦 2 − 8𝑥𝑦 2 + 2𝑥𝑦 + 4𝑥𝑦 = 3𝑧 4 − 6𝑥𝑦 2 + 6𝑥𝑦 Let 𝑓(𝑥) = 10𝑥 2 − 3𝑥 + 8 and 𝑔(𝑥) = 4𝑥 2 + 8𝑥 − 9 Solution: 𝑓(𝑥) + 𝑔(𝑥) is asking to add the right side of the functions in the specific order. Find 𝑓(𝑥) + 𝑔(𝑥) And g(𝑥) − 𝑓(𝑥) 𝑓(𝑥) + 𝑔(𝑥) = (10𝑥 2 − 3𝑥 + 8) + (4𝑥 2 + 8𝑥 − 9) = 10𝑥 2 − 3𝑥 + 8 + 4𝑥 2 + 8𝑥 − 9 = 10𝑥 2 + 4𝑥 2 − 3𝑥 + 8𝑥 + 8 − 9 Answer: 𝑓(𝑥) + 𝑔(𝑥) = 14𝑥 2 − 5𝑥 − 1 g(𝑥) − 𝑓(𝑥) is asking to subtract the right side of the functions in the specific order. 𝑔(𝑥) − 𝑓(𝑥) = (4𝑥 2 + 8𝑥 − 9) − (10𝑥 2 − 3𝑥 + 8) = 4𝑥 2 + 8𝑥 − 9 − 10𝑥 2 + 3𝑥 − 8 =4𝑥 2 − 10𝑥 2 + 8𝑥 + 3𝑥 − 9 − 8 Answer: 𝑔(𝑥) − 𝑓(𝑥) = −6𝑥 2 + 11𝑥 − 17 NOTE: (𝑓 + 𝑔)(𝑥) = 𝑓(𝑥) + 𝑔(𝑥) and (𝑓 − 𝑔)(𝑥) = 𝑓(𝑥) − 𝑔(𝑥) Let 𝑓(𝑥) = 8𝑥 + 5 and 𝑔(𝑥) = 9𝑥 − 3 Find: ℎ(𝑥) = (𝑓 + 𝑔)(𝑥) 𝑘(𝑥) = (𝑔 − 𝑓)(𝑥) Solution: ℎ(𝑥) = (𝑓 + 𝑔)(𝑥) = 𝑓(𝑥) + 𝑔(𝑥) = (8𝑥 + 5) + (9𝑥 − 3) = 8𝑥 + 5 + 9𝑥 − 3 Solution: ℎ(𝑥) = 17𝑥 + 2 𝑘(𝑥) = (𝑔 − 𝑓)(𝑥) = 𝑔(𝑥) − 𝑓(𝑥) = (9𝑥 − 3) − (8𝑥 + 5) = 9𝑥 − 3 − 8𝑥 − 5 Solution: 𝑥 − 8 Section 4.1: Polynomial Addition and subtraction 𝑙𝑒𝑡 𝑓(𝑥) = 𝑥 2 − 9𝑥 + 3 and 𝑔(𝑥) = 12𝑥 − 5 Find: 𝑓(3) + 𝑔(5) And 𝑔(6) − 𝑓(4) Chapter 4: Polynomial and Radical Operations Replace the x in the 𝑓 − 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛 with 3, and the x in the 𝑔 − 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛 with 5 then simplify and add the results. 𝑓(3) + 𝑔(5) 𝑓(3) + 𝑔(5) = ((3)2 − 9(3) + 3) + (12(5) − 5) = (9 − 9(3) + 3) + (60 − 5) = (9 − 27 + 3) + (60 − 5) = −15 + 55 Answer: 𝑓(3) + 𝑔(5) = 40 𝑔(6) − 𝑓(4) Replace the x in the 𝑔 − 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛 with 6, and the x in the 𝑓 − 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛 with 4 then simplify and subtract the results. 𝑔(6) − 𝑓(4) = (12(6) − 5) − ((4)2 − 9(4) + 3) = (72 − 5) − (16 − 36 + 3) = 67 − 17 Solution: 50