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7-5 Polynomials A monomial is a number, a variable, or a product of numbers and variables with whole-number exponents. The degree of a monomial is the sum of the exponents of the variables. A constant has degree 0. Holt Algebra 1 7-5 Polynomials Example : Finding the Degree of a Monomial Find the degree of each monomial. A. 4p4q3 The degree is 7. B. 7ed The degree is 2. C. 3 The degree is 0. Holt Algebra 1 Add the exponents of the variables: 4 + 3 = 7. Add the exponents of the variables: 1+ 1 = 2. Add the exponents of the variables: 0 = 0. 7-5 Polynomials A polynomial is a monomial or a sum or difference of monomials. The degree of a polynomial is the degree of the term with the greatest degree. Holt Algebra 1 7-5 Polynomials Example : Finding the Degree of a Polynomial Find the degree of each polynomial. A. 11x7 + 3x3 11x7: degree 7 3x3: degree 3 The degree of the polynomial is the greatest degree, 7. Find the degree of each term. B. :degree 3 –5: degree 0 :degree 4 Find the degree of each term. The degree of the polynomial is the greatest degree, 4. Holt Algebra 1 7-5 Polynomials Check It Out! Find the degree of each polynomial. a. 5x – 6 5x: degree 1 –6: degree 0 The degree of the polynomial is the greatest degree, 1. Find the degree of each term. b. x3y2 + x2y3 – x4 + 2 x3y2: degree 5 –x4: degree 4 x2y3: degree 5 2: degree 0 The degree of the polynomial is the greatest degree, 5. Holt Algebra 1 Find the degree of each term. 7-5 Polynomials The terms of a polynomial may be written in any order. However, polynomials that contain only one variable are usually written in standard form. The standard form of a polynomial that contains one variable is written with the terms in order from greatest degree to least degree. When written in standard form, the coefficient of the first term is called the leading coefficient. Holt Algebra 1 7-5 Polynomials Example : Writing Polynomials in Standard Form Write the polynomial in standard form. Then give the leading coefficient. 6x – 7x5 + 4x2 + 9 Find the degree of each term. Then arrange them in descending order: 6x – 7x5 + 4x2 + 9 Degree 1 5 2 0 –7x5 + 4x2 + 6x + 9 5 2 1 0 The standard form is –7x5 + 4x2 + 6x + 9. The leading coefficient is –7. Holt Algebra 1 7-5 Polynomials Check It Out! Write the polynomial in standard form. Then give the leading coefficient. 16 – 4x2 + x5 + 9x3 Find the degree of each term. Then arrange them in descending order: 16 – 4x2 + x5 + 9x3 Degree 0 2 5 3 x5 + 9x3 – 4x2 + 16 5 3 2 0 The standard form is x5 + 9x3 – 4x2 + 16. The leading coefficient is 1. Holt Algebra 1 7-5 Polynomials Some polynomials have special names based on their degree and the number of terms they have. Degree Name Terms Name 0 Constant 1 Monomial 1 Linear 2 Binomial 2 Quadratic Trinomial 3 4 Cubic Quartic 3 4 or more 5 Quintic 6 or more Holt Algebra 1 6th,7th,degree and so on Polynomial 7-5 Polynomials Example: Classifying Polynomials Classify each polynomial according to its degree and number of terms. A. 5n3 + 4n Degree 3 Terms 2 5n3 + 4n is a cubic binomial. B. 4y6 – 5y3 + 2y – 9 Degree 6 Terms 4 4y6 – 5y3 + 2y – 9 is a C. –2x Degree 1 Terms 1 –2x is a linear monomial. Holt Algebra 1 6th-degree polynomial. 7-5 Polynomials Check It Out! Classify each polynomial according to its degree and number of terms. a. x3 + x2 – x + 2 Degree 3 Terms 4 x3 + x2 – x + 2 is a cubic polymial. b. 6 Degree 0 Terms 1 6 is a constant monomial. c. –3y8 + 18y5 + 14y Degree 8 Terms 3 –3y8 + 18y5 + 14y is an 8th-degree trinomial. Holt Algebra 1 7-5 Polynomials Just as you can perform operations on numbers, you can perform operations on polynomials. To add or subtract polynomials, combine like terms. Holt Algebra 1 7-5 Polynomials Example: Adding and Subtracting Monomials Add or Subtract.. A. 12p3 + 11p2 + 8p3 12p3 + 11p2 + 8p3 12p3 + 8p3 + 11p2 20p3 + 11p2 B. 5x2 – 6 – 3x + 8 5x2 – 6 – 3x + 8 5x2 – 3x + 8 – 6 5x2 – 3x + 2 Holt Algebra 1 Identify like terms. Rearrange terms so that like terms are together. Combine like terms. Identify like terms. Rearrange terms so that like terms are together. Combine like terms. 7-5 Polynomials Remember! Like terms are constants or terms with the same variable(s) raised to the same power(s). To review combining like terms, see lesson 1-7. Holt Algebra 1 7-5 Polynomials Check It Out! Add or subtract. a. 2x8 + 7y8 – x8 – y8 2x8 7y8 + – – 2x8 – x8 + 7y8 – y8 x8 + 6y8 Holt Algebra 1 x8 y8 Identify like terms. Rearrange terms so that like terms are together. Combine like terms. 7-5 Polynomials Polynomials can be added in either vertical or horizontal form. In vertical form, align the like terms and add: 5x2 + 4x + 1 + 2x2 + 5x + 2 7x2 + 9x + 3 In horizontal form, use the Associative and Commutative Properties to regroup and combine like terms. (5x2 + 4x + 1) + (2x2 + 5x + 2) = (5x2 + 2x2 + 1) + (4x + 5x) + (1 + 2) = 7x2 + 9x + 3 Holt Algebra 1 7-5 Polynomials Example: Adding Polynomials Add. A. (4m2 + 5) + (m2 – m + 6) (4m2 + 5) + (m2 – m + 6) Identify like terms. (4m2 + m2) + (–m) +(5 + 6) Group like terms together. Combine like terms. 5m2 – m + 11 B. (10xy + x) + (–3xy + y) (10xy + x) + (–3xy + y) Identify like terms. (10xy – 3xy) + x + y Group like terms together. Combine like terms. 7xy + x + y Holt Algebra 1 7-5 Polynomials To subtract polynomials, remember that subtracting is the same as adding the opposite. To find the opposite of a polynomial, you must write the opposite of each term in the polynomial: –(2x3 – 3x + 7)= –2x3 + 3x – 7 Holt Algebra 1 7-5 Polynomials Example: Subtracting Polynomials Subtract. (7m4 – 2m2) – (5m4 – 5m2 + 8) (7m4 – 2m2) + (–5m4 + 5m2 – 8) Rewrite subtraction as addition of the opposite. (7m4 – 2m2) + (–5m4 + 5m2 – 8) Identify like terms. (7m4 – 5m4) + (–2m2 + 5m2) – 8 Group like terms together. 2m4 + 3m2 – 8 Holt Algebra 1 Combine like terms. 7-5 Polynomials Example: Subtracting Polynomials Subtract. (–10x2 – 3x + 7) – (x2 – 9) (–10x2 – 3x + 7) + (–x2 + 9) (–10x2 – 3x + 7) + (–x2 + 9) –10x2 – 3x + 7 –x2 + 0x + 9 –11x2 – 3x + 16 Holt Algebra 1 Rewrite subtraction as addition of the opposite. Identify like terms. Use the vertical method. Write 0x as a placeholder. Combine like terms. 7-5 Polynomials Check It Out! Subtract. (2x2 – 3x2 + 1) – (x2 + x + 1) (2x2 – 3x2 + 1) + (–x2 – x – 1) Rewrite subtraction as addition of the opposite. (2x2 – 3x2 + 1) + (–x2 – x – 1) Identify like terms. –x2 + 0x + 1 + –x2 – x – 1 –2x2 – x Holt Algebra 1 Use the vertical method. Write 0x as a placeholder. Combine like terms. 7-5 Polynomials Lesson Quiz: Part I Find the degree of each polynomial. 1. 7a3b2 – 2a4 + 4b – 15 2. 25x2 – 3x4 5 4 Write each polynomial in standard form. Then give the leading coefficient. 3. 24g3 + 10 + 7g5 – g2 7g5 + 24g3 – g2 + 10; 7 4. 14 – x4 + 3x2 Holt Algebra 1 –x4 + 3x2 + 14; –1 7-5 Polynomials Lesson Quiz: Part II Classify each polynomial according to its degree and number of terms. 5. 18x2 – 12x + 5 6. 2x4 – 1 quadratic trinomial quartic binomial Add or subtract. 7. 7m2 + 3m + 4m2 11m2 + 3m 8. (r2 + s2) – (5r2 + 4s2) (–4r2 – 3s2) 9. (10pq + 3p) + (2pq – 5p + 6pq) 10. (14d2 + 1) + (6d2 – 2d - 8) Holt Algebra 1 18pq – 2p 20d2 – 2d – 7 7-5 Polynomials Lesson Quiz: Part III Add or subtract. 11. 7m2 + 3m + 4m2 11m2 + 3m 12. (r2 + s2) – (5r2 + 4s2) (–4r2 – 3s2) 13. (10pq + 3p) + (2pq – 5p + 6pq) 18pq – 2p 14. (14d2 + 1) + (6d2 – 2d - 8) 15. (2ab + 14b) – (–5ab + 4b) Holt Algebra 1 20d2 – 2d – 7 7ab + 10b