1. 1 Definition of Monomials: Problems STANDARD 10 Definition of Polynomials: Problems POLYNOMIALS Standard Form of a Polynomial: Problems Adding Polynomials: Vertical and Horizontal Formats, problems Adding Polynomials when a term is missing: Problems Subtracting Polynomials: Vertical Format, problems Subtracting Polynomials: Horizontal Format, problems Adding Polynomials with more than one variable, problems END SHOW PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

2. 2 STANDARD 10: Students add, subtract, multiply, and divide monomials and polynomials. Students solve multi-step problems, including word problems, by using these techniques. ESTÁNDAR 10: Los estudiantes suman restan, multiplican, y dividen monomios y polinomios. Los estudiantes resuelven problemas de múltiples pasos, incluyendo problemas escritos, usando estas técnicas. PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

3. 3 STANDARD 10 MONOMIALS: 9 4x 3pq 3 1 4 xy These are examples of monomials: A monomial is a number, a variable, or the product of a number and one or more variables with whole number exponents. The degree of a monomial is the sum of the exponents of the variables in the monomial. Find the degree of the following monomials: x 6 r t 46 x y z 3 54 2 2 20 6 4 + 6 = 10 3 + 5 +4 = 12 = 20x 0 0 x y z 2 2 1 = 2 + 1 + 2 = 5 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

4. 4 STANDARD 10 POLYNOMIALS: A polynomial is a monomial or a sum of monomials. The degree of a polynomial in one variable is the largest exponent of that variable. Polynomials are also identified by number of terms. POLYNOMIAL DEGREE IDENTIFIED BY DEGREE IDENTIFIED BY NUMBER OF TERMS 5 0 constantmonomial 1 6y +3 1 linear binomial x - 3x + 9 2 2 quadratic trinomial 4z – 9z - 2 23 3 cubic trinomial PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

5. 5 STANDARD 10 POLYNOMIALS: Polynomials are most commonly written in STANDARD FORM, that is in decreasing order, from largest exponent to smallest exponent. Write the following polynomials in STANDARD FORM: -3x + x - 2 2 2 1 x 0 x -3x -2 2 3x – x + 4x + 1 - x 2 3 4 -x - x + 3x + 4x - 1 4 3 2 1 4 3 2 6x + 2x -x + 2 5 3 2 x 4 0 x 1 0 2x + + 6x – x + + 2 5 3 2 x 0 2x + 6x – x + 2 5 3 2 8x + 5x + 3x -x + 7 6 3 4 8x + + 3x + 5x + - x + 7 6 4 3 x 5 0 x 2 0 1 8x + 3x + 5x - x + 7 6 4 3 x 0 x 0 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

6. 6 STANDARD 10 ADDING POLYNOMIALS: VERTICAL FORMAT: Find the sums of the following polynomial: (4x + 5x -3) + (3x -x + 10) 2 2 2 4x + 5x - 3 2 3x - x + 10 1 + 7 + 4x 7x 2 HORIZONTAL FORMAT: 2 4x + 5x - 3 + 2 3x - x + 10 = 4x + 3x + 5x -1x -3 +10 2 2 = 7x + 4x + 7 2 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

7. 7 STANDARD 10 ADDING POLYNOMIALS: VERTICAL FORMAT: Find the sums of the following polynomial: (5x + 2x -4) + (4x +7x + 12) 2 2 2 5x + 2x - 4 2 4x + 7x + 12 + 8+ 9x 9x 2 HORIZONTAL FORMAT: 2 5x + 2x - 4 + 2 4x +7x + 12 = 5x + 4x + 2x +7x -4 +12 2 2 = 9x + 9x + 8 2 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

8. 8 STANDARD 10 ADDING POLYNOMIALS: Find the sums of the following polynomials using VERTICAL FORMAT: (3x - 5) + (9x + 2x + 11) 2 2 2 9x + 2x +11 3x + 0x - 5 2 + 6+ 2x 12x 2 (4x + 7x + x + 3) + ( 2x - x + 4 – 3x ) 4 3 2 3 4 4x + 7x + 1x + 0x + 3 4 3 2 -3x + 2x + 0x - 1x + 4 4 3 2 + 7 - x + x 2 + 9x 3 x 4 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

9. 9 STANDARD 10 ADDING POLYNOMIALS: Find the sums of the following polynomials using HORIZONTAL FORMAT: (3x - 5) + (9x + 2x + 11) 2 2 2 9x + 2x +11 3x - 5 2 (4x + 7x + x + 3) + ( 2x - x + 4 – 3x ) 4 3 2 3 4 4x + 7x + x + 3 4 3 2 -3x + 2x -x + 4 4 3 + = 3x + 9x + 2x – 5 + 11 22 + 6+ 2x 12x 2 = + + 7 - x + x 2 + 9x 3 x 4 = = 4x – 3x +7x +2x + x -x + 3 + 4 4 4 3 3 2 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

10. 10 STANDARD 10 SUBTRACTING POLYNOMIALS: Find the difference of the following polynomials using VERTICAL FORMAT: (6x - 2) - (8x + 4x + 14) 2 2 2 8x + 4x +14 6x + 0x - 2 2 -16- 4x -2x 2 (9x + 5x + 2x + 5) - ( 4x - 2x + 7 – 6x ) 4 3 2 3 4 9x + 5x + 2x + 0x + 5 4 3 2 -6x + 4x + 0x - 2x + 7 4 3 2 - 2 +2x 2 + x 3 15x 4 - 2 -8x - 4x -14 6x + 0x - 2 2 - 9x + 5x + 2x + 0x + 5 4 3 2 6x - 4x - 0x + 2x - 7 4 3 2 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

11. 11 STANDARD 10 SUBTRACTING POLYNOMIALS: Find the differences of the following polynomials using HORIZONTAL FORMAT: (4x - 5) - (8x + 3x + 10) 2 2 2 8x + 3x +10 4x - 5 2 (5x + 6x + x + 4) - ( 3x - x + 5 – 2x ) 4 3 2 3 4 5x + 6x + x + 4 4 3 2 -2x + 3x -x + 5 4 3 = 4x - 8x - 3x – 5 - 10 22 - 15- 3x -4x 2 = - 1 + x 2 + 3x 3 7x 4 = = 5x + 2x + 6x - 3x + x + x + 4 - 5 4 4 3 3 2 - 2 8x - 3x -10 4x - 5 2 = - - 5x + 6x + x + 4 4 3 2 +2x - 3x + x - 5 4 3 = PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

12. 12 It is possible to add or subtract terms of a polynomial only if they are LIKE TERMS: Simplify 5xy + 6z x -9xy + 10z x – 15z 5 5 3 5xy + 6z x -9xy + 10z x – 15z 5 5 3 Simplify -8a b c + 7b c - 3b c + a b c 3 5 3 5 3 5 2 5 -8a b c + 7b c - 3b c + a b c 3 5 3 5 3 5 2 5 = -8a b c + a b c + 7b c – 3b c 3 5 3 5 3 5 2 5 = -7a b c + 7b c -3b c 3 5 3 5 2 5 = -7a b c - 3b c +7b c 3 5 2 5 3 5 = 5xy – 9xy + 6z x + 10z x -15z 5 5 3 = -4xy +16z x – 15z 5 3 STANDARD 10 ADDING POLYNOMIALS: PRESENTATION CREATED BY SIMON PEREZ. All rights reserved