Polynomials Monomial - a Number, a Variable or a PRODUCT of a number and a variable. Monomials cannot have radicals with variables inside, quotients of variables or variables with negative exponents. Degree of a monomial - is the SUM of the exponents of the variable(s) in the monomial. The degree of a constant term is 0. Polynomials a variable expression whose terms are Monomials. Monomials have 1 term. Binomials have 2 terms. Trinomials have 3 terms. Polynomials with more than 3 terms do not have special names. Polynomials in one variable are usually arranged in descending order so that the exponents of the variables decrease from left to right. Polynomials (just like monomials) cannot have radicals with variables inside, quotients of variables or variables with negative exponents. Degree of Polynomial is the Greatest of the degrees of any of its terms. (Remember each term is a monomial so the degree will be the sum of the exponents in the monomial.) Constant Term of Polynomial Term that does not have a variable attached to it. Leading Coefficient of Polynomial is the coefficient of the variable with the largest exponent. What Can You Do With Polynomials? Evaluate Polynomials Just substitute in the assigned value for the variable and find the value of the polynomial. Example: 3x 2 + 4x + 6 evaluatex = 3. The value of the polynomial will be 45. 3(3) 2 + 4(3) + 6 = 45. Add Polynomials To add polynomials just combine like terms. There are 2 formats you can use to add polynomials horizontal format or vertical format. Example: Horizontal Format: (3x 2 + 4x + 6) + (7x 2 + 2x 2) = 10x2 + 6x + 4 Vertical Format (3x 2 + 4x + 6) + (7x 2 + 2x 2) 10x2 + 6x + 4 Subtract Polynomials To subtract polynomials just add the additive inverse of the 2 nd polynomial. There are 2 formats you can use to subtract polynomials horizontal format or vertical format. Example: Horizontal Format: (3x 2 + 4x + 6) (7x 2 + 2x 2) = 4x 2 + 2x + 8 Vertical Format (3x 2 + 4x + 6) (7x 2 + 2x 2) 4x 2 + 2x + 8 HighSchoolMathTeachers 2018 Page 1
Part 1: Find the degree of the polynomial. 1. 3x + 2 2. 4x 3 + 17x 5 3. 13x n + 5x n 2 4. 0.55x 2 + 0.7x 5. 7x 3 + 10x + 15 Part 2: Standard form of a polynomial is when the terms are written so that the exponents on the variables decrease from left to right. Write the polynomial in standard form. 6. 3x 2 + x + 3x 3 12 7. 4 + x 3 3x 2 8. 4x + 5 + 2x 2 9. 7 + 6x + 3x 2 + x 3 Like terms in a polynomial have the same and the same. Part 3: Identify the like terms and simplify the expression. 10. x 2 + 5x 3 2x + 3x 2 + 7 4x + 7 11. 6 + x 2x 2 + 5 + x 2 12. 2x 2 5x 3 + x 2 18 + 5x 3x 3 13. 15x 8 + 4x 3 + 6x 15 x 7 + 2x Part 4: Leading Coefficient and Constant Term. Identify the leading coefficient in each of the polynomials in part 3. 10. 11. 12. 13. Identify the constant term in each of the polynomials in part 3. 10. 11. 12. 13. HighSchoolMathTeachers 2018 Page 2
Adding and Subtracting Polynomials To add polynomials simply combine like terms!! Horizontal Format: 1. (3x 2 + 2x 7) + (7x 3 3 + 4x 2 ) = 2. (4x 2 + 5x 3) + (7x 3 7x + 1) = Vertical Format: 3. (2x 3x 2 + 4x 3 + 1) + (3x 2 2x + 7) 4. (5x 4 2x + x 2 + 4) + ( 5 + 6x) HighSchoolMathTeachers 2018 Page 3
To subtract polynomials distribute the negative and then add!! (*Think subtraction means add the opposite*) Horizontal Format: 5. (3x 2 7xy + y 2 ) ( 4x 2 + 7xy 3y 2 ) 6. (6x 3 3x + 7) (3x 2 5x + 12) Vertical Format: 7. (3x 2 2x + 5) (2 3x + 4x 3 ) 8. (5x 2 + 3x 1) (2x 2 + 4x 6) HighSchoolMathTeachers 2018 Page 4
Unit 3 ~ Day A HW Complete the chart below by first rewriting each polynomial in standard form. Then identify the degree, leading coefficient, and constant for each polynomial. Rewrite in Standard Polynomial Degree Form Leading Coefficient Constant 1. 2 + 5x 2. 12 3. 1 4x 2 + 7x 4. 5x 3 2 5. 4 3x 4 6. x 2 + 5x + 2x 4 x + 1 7. 3x 3 10x + 15 + 6x 5 8. 7x x 5 + 3x 2 + 3x 5 9. 6 3x 3 10. 5x + 14x 4 8 4x + 2x 2 HighSchoolMathTeachers 2018 Page 5
Check the box (as) that describes each polynomial below. Monomial Binomial Trinomial Polynomial 11. x 2 5x + 7 12. 10x 4 13. 3 15x 3 14. 10x 5 12x 3 + 8 15. 6x 9 10x 5 + 2x 19 Add or subtract each polynomial. Choose a method (horizontal/vertical): 16. (4x 2 + 3x 5) + (x 2 7x + 10) 17. (x 2 2x + 7) (3x 2 4x + 7) 18. (4a 2 7a) ( 6a 2 + 5a 7) 19. (5x 2 3x + 8) (x 2 8x + 12) 20. (3x 2 2x + 7) + ( 3x 2 + 2x 12) 21. (5x 2 x + 6) + ( 3x 2 + 2x 12) + ( 6x 2 + 5x 7) HighSchoolMathTeachers 2018 Page 6
Answer Key Part 1 1. 1 2. 3 3. N 4. 2 5. 3 Part 2 6. 3x 3 + 3x 2 + x 12 7. x 3 3x 2 + 4 8. 2x 2 4x + 5 9. x 3 + 3x 2 + 6x 7 Like terms in a polynomial have the same _Variable_ and the same _Degree_. Part 3 10. 5x 3 + 4x 2 6x + 14 11. x 2 + x + 11 12. 8x 3 + 3x 2 + 5x 18 13. 15x 8 x 7 + 4x 3 + 8x 15 Part 4 10. 5 11. -1 12. -8 13. 15 10. 14 11. 11 12. -18 13. -15 Adding and Subtracting Polynomials 1. 7x 3 + 7x 2 + 2x 10 2. 7x 3 + 4x 2 2x 2 3. 4x 3 + 8 4. 5x 4 + x 2 + 4x 1 5. 7x 2 14xy + 4y 2 6. 6x 3 3x 2 + 2x 5 7. 4x 3 + 3x 2 + x + 3 8. 3x 2 x + 5 HighSchoolMathTeachers 2018 Page 7
Unit 3 Polynomial Rewrite in Standard Form Degree Leading Coefficient Constant 1. 2 + 5x 5x 2 1 5-2 2. 12 12 0 12 3. 1 4x 2 + 7x 4x 2 + 7x 1 2-4 -1 4. 5x 3 2 5x 3 2 3 5-2 5. 4 3x 4 3x 4 + 4 4-3 4 6. x 2 + 5x + 2x 4 x + 1 2x 4 + x 2 + 4x + 1 4 2 1 7. 3x 3 10x + 15 + 6x 5 6x 5 + 3x 3 10x + 15 5 6 15 8. 7x x 5 + 3x 2 + 3x 5 2x 5 + 3x 2 + 7x 5 2 0 9. 6 3x 3 3x 3 + 6 3-3 6 10. 5x + 14x 4 8 4x + 2x 2 14x 4 + 2x 2 + x 8 4 14-8 HighSchoolMathTeachers 2018 Page 8
Monomial Binomial Trinomial Polynomial 11. x 2 5x + 7 x 12. 10x 4 x 13. 3 15x 3 x 14. 10x 5 12x 3 + 8 x 15. 6x 9 10x 5 + 2x 19 x Add or subtract each polynomial. 16. 5x 2 4x + 5 17. 2x 2x 2 18. 10a 2 12a + 7 19. 4x 2 + 5x 4 20. 5 21. 4x 2 + 6x 13 HighSchoolMathTeachers 2018 Page 9