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 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 STANDARD 10 MONOMIALS: 9 4x 3pq 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 = = 12 = 20x 0 0 x y z = = 5 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

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 quadratic trinomial 4z – 9z cubic trinomial PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

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 x 0 x -3x x – x + 4x x x - x + 3x + 4x x + 2x -x x 4 0 x 1 0 2x + + 6x – x x 0 2x + 6x – x x + 5x + 3x -x x + + 3x + 5x + - x x 5 0 x x + 3x + 5x - x x 0 x 0 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

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

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

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

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

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

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

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 xy + 6z x -9xy + 10z x – 15z Simplify -8a b c + 7b c - 3b c + a b c a b c + 7b c - 3b c + a b c = -8a b c + a b c + 7b c – 3b c = -7a b c + 7b c -3b c = -7a b c - 3b c +7b c = 5xy – 9xy + 6z x + 10z x -15z = -4xy +16z x – 15z 5 3 STANDARD 10 ADDING POLYNOMIALS: PRESENTATION CREATED BY SIMON PEREZ. All rights reserved