Slide 1 / 211 New Jersey Center for Teaching and Learning Progressive Mathematics Initiative This material is made freely available at www.njctl.org and is intended for the non-commercial use of students and teachers. These materials may not be used for any commercial purpose without the written permission of the owners. NJCTL maintains its website for the convenience of teachers who wish to make their work available to other teachers, participate in a virtual professional learning community, and/or provide access to course materials to parents, students and others. Click to go to website: www.njctl.org Slide 2 / 211 Algebra I Polynomials 2013-07-31 www.njctl.org Slide 3 / 211 Table of Contents · Definitions of Monomials, Polynomials and Degrees · Adding and Subtracting Polynomials · Mulitplying a Polynomial by a Monomial · Multiplying Polynomials · Special Binomial Products · Solving Equations · Factors and GCF · Factoring out GCF's · Identifying & Factoring x 2 + bx + c · Factoring Using Special Patterns · Factoring Trinomials ax 2 + bx + c · Factoring 4 Term Polynomials · Mixed Factoring · Solving Equations by Factoring
Slide 4 / 211 Definitions of Monomials, Polynomials and Degrees Return to Table of Contents Slide 5 / 211 A monomial is a one- term expression formed by a number, a variable, or the product of numbers and variables. 2 4x 8 Examples of monomials.... 81y 4 z rt 32,457 2 6 17x n 3 m Slide 6 / 211 Drag the following terms into the correct sorting box. If you sort correctly, the term will be visible. If you sort incorrectly, the term will x 2 (5 + 7y) t 7x - 12 3 disappear. y 5 - 4 6 1 a + b - 5 6+ 5rs 5 x + 7 48x 2 yz 3 xy 4 2 ) 15 b c 2 5 a 4 ( 7 Monomials
Slide 7 / 211 A polynomial is an expression that contains two or more monomials. c 2 + d Examples of polynomials... . 7+ b+ c 5+ a 2 2 + 4d 8a 3 - 2b 2 3 2 x + 4c- mn 3 rt a 4 b 3 x 8 + 15 6 Slide 8 / 211 Degrees of Monomials The degree of a monomial is the sum of the exponents of its variables. The degree of a nonzero constant such as 5 or 12 is 0. The constant 0 has no degree. Examples: 1) The degree of 3x is? 1 The variable x has a degree 1. 2) The degree of -6x 3 y is? 4 The x has a power of 3 and the y has a power of 1, so the degree is 3+1 =4. 3) The degree of 9 is? 0 A constant has a degree 0, because there is no variable. Slide 9 / 211 1 What is the degree of ? 0 A Pull Pull B 1 C 2 3 D
Slide 10 / 211 2 What is the degree of ? A 0 Pull Pull 1 B 2 C 3 D Slide 11 / 211 3 What is the degree of 3 ? 0 A Pull Pull 1 B 2 C D 3 Slide 12 / 211 4 What is the degree of ? Pull Pull
Slide 13 / 211 Degrees of Polynomials The degree of a polynomial is the same as that of the term with the greatest degree. Example: Find degree of the polynomial 4x 3 y 2 - 6xy 2 + xy. The monomial 4x 3 y 2 has a degree of 5, the monomial 6xy 2 has a degree of 3, and the monomial xy has a degree of 2. The highest degree is 5, so the degree of the polynomial is 5. Slide 14 / 211 Find the degree of each polynomial Answers: 1) 0 1) 3 2) 3 2) 12c 3 3) 2 3) ab 4) 5 4) 8s 4 t 5) 2 - 7n 5) 1 6) 4 6) h 4 - 8t 7) 4 7) s 3 + 2v 2 y 2 - 1 Slide 15 / 211 5 What is the degree of the following polynomial: 3 A B 4 Pull Pull C 5 6 D
Slide 16 / 211 6 What is the degree of the following polynomial: A 3 Pull Pull 4 B 5 C 6 D Slide 17 / 211 7 What is the degree of the following polynomial: 3 A Pull Pull 4 B 5 C D 6 Slide 18 / 211 8 What is the degree of the following polynomial: 3 A B 4 Pull Pull C 5 6 D
Slide 19 / 211 Adding and Subtracting Polynomials Return to Table of Contents Slide 20 / 211 Standard Form The standard form of an equation is to put the terms in order from highest degree to the lowest degree. Standard form is commonly excepted way to write polynomials. Example: is in standard form. Put the following equation into standard form: Slide 21 / 211 Monomials with the same variables and the same power are like terms. Like Terms Unlike Terms 4x and -12x -3b and 3a x 3 y and 4x 3 y 6a 2 b and -2ab 2
Slide 22 / 211 Combine these like terms using the indicated operation. Slide 23 / 211 9 Simplify A Pull Pull B C D Slide 24 / 211 10 Simplify A Pull Pull B C D
Slide 25 / 211 11 Simplify A Pull Pull B C D Slide 26 / 211 To add polynomials, combine the like terms from each polynomial. To add vertically, first line up the like terms and then add. Examples: (3x 2 +5x -12) + (5x 2 -7x +3) (3x 4 -5x) + (7x 4 +5x 2 -14x) line up the like terms line up the like terms 3x 2 + 5x - 12 3x 4 -5x (+)5x 2 - 7x + 3 (+) 7x 4 +5x 2 - 14x 8x 2 - 2x - 9 10x 4 +5x 2 - 19x = Slide 27 / 211 We can also add polynomials horizontally. (3x 2 + 12x - 5) + (5x 2 - 7x - 9) Use the communitive and associative properties to group like terms. (3x 2 + 5x 2 ) + (12x + -7x) + (-5 + -9) 8x 2 + 5x - 14
Slide 28 / 211 12 Add A Pull Pull B C D Slide 29 / 211 13 Add A Pull Pull B C D Slide 30 / 211 14 Add A Pull Pull B C D
Slide 31 / 211 15 Add A Pull Pull B C D Slide 32 / 211 16 Add A Pull Pull B C D Slide 33 / 211 To subtract polynomials, subtract the coefficients of like terms. Example: -3x - 4x = -7x 13y - (-9y) = 22y 6xy - 13xy = -7xy
Slide 34 / 211 We can subtract polynomials vertically and horizontally. To subtract a polynomial, change the subtraction to adding -1. Distribute the -1 and then follow the rules for adding polynomials (3x 2 +4x -5) - (5x 2 -6x +3) (3x 2 +4x-5) +(-1) (5x 2 -6x+3) (3x 2 +4x-5) + (-5x 2 +6x-3) 3x 2 + 4x - 5 (+) -5x 2 - 6x + 3 -2x 2 +10x - 8 Slide 35 / 211 We can subtract polynomials vertically and horizontally. To subtract a polynomial, change the subtraction to adding -1. Distribute the -1 and then follow the rules for adding polynomials (4x 3 -3x -5) - (2x 3 +4x 2 -7) (4x 3 -3x -5) +(-1)(2x 3 +4x 2 -7) (4x 3 -3x -5) + (-2x 3 -4x 2 +7) 4x 3 - 3x - 5 (+) -2x 3 - 4x 2 + 7 2x 3 - 4x 2 - 3x + 2 Slide 36 / 211 We can also subtract polynomials horizontally. (3x 2 + 12x - 5) - (5x 2 - 7x - 9) Change the subtraction to adding a negative one and distribute the negative one. (3x 2 + 12x - 5) +(-1)(5x 2 - 7x - 9) (3x 2 + 12x - 5) + (-5x 2 + 7x + 9) Use the communitive and associative properties to group like terms. (3x 2 +-5x 2 ) + (12x +7x) + (-5 +9) -2x 2 + 19x + 4
Slide 37 / 211 17 Subtract A Pull Pull B C D Slide 38 / 211 18 Subtract A Pull Pull B C D Slide 39 / 211 19 Subtract A Pull Pull B C D
Slide 40 / 211 20 Subtract A Pull Pull B C D Slide 41 / 211 21 Subtract A Pull Pull B C D Slide 42 / 211 22 What is the perimeter of the following figure? (answers are in units) Pull Pull A B C D
Slide 43 / 211 Multiplying a Polynomial by a Monomial Return to Table of Contents Slide 44 / 211 Find the total area of the rectangles. 3 5 8 4 square units square units Slide 45 / 211 To multiply a polynomial by a monomial, you use the distributive property together with the laws of exponents for multiplication. Examples: Simplify. - 2x (5x 2 - 6x + 8) - 2x (5x 2 + - 6x + 8) (- 2x )(5x 2 ) + ( - 2x )(- 6x) + ( - 2x )(8) 3 + 12x 2 + - 16x - 10x - 10x 3 + 12x 2 - 16x
Slide 46 / 211 To multiply a polynomial by a monomial, you use the distributive property together with the laws of exponents for multiplication. Examples: Simplify. - 3x 2 (- 2x 2 + 3x - 12) - 3x 2 (- 2x 2 + 3x + - 12) (- 3x 2 )(- 2x 2 ) + ( - 3x 2 )(3x) + ( - 3x 2 )(- 12) 6x 4 + - 9x 3 + 36x 2 6x 4 - 9x 3 + 36x 2 Slide 47 / 211 Let's Try It! Multiply to simplify. 1. - 2x 4 + 4x 3 - 7x 2 Slide to check. Slide to check. 2. 4x 2 (5x 2 - 6x - 3) 20x 4 - 24x 3 - 12x Slide to check. 3. 3xy(4x 3 y 2 - 5x 2 y 3 + 8xy 4 ) 12x 4 y 3 - 15x 3 y 4 + 24x 2 y 5 Slide 48 / 211 23 What is the area of the rectangle shown? A Pull Pull x 2 + 2x + 4 B x 2 C D
Slide 49 / 211 24 A 6x 2 + 8x - 12 Pull Pull 6x 2 + 8x 2 - 12 B 6x 2 + 8x 2 - 12x C 6x 3 + 8x 2 - 12x D Slide 50 / 211 25 A Pull Pull B C D Slide 51 / 211 26 A Pull Pull B C D
Slide 52 / 211 27 Find the area of a triangle (A= 1 / 2 bh) with a base of 4x and a height of 2x - 8. All answers are in square units. A Pull Pull B C D Slide 53 / 211 Multiplying Polynomials Return to Table of Contents Slide 54 / 211 Find the total area of the rectangles. 8 5 2 6 (2 + 6) (5 + 8 ) Area of the big rectangle = 2 (5 + 8) + 6 (5 + 8) Area of the horizontal rectangles Area of each box = 2(5) + 2(8) + 6(5) + 6(8) = 10 + 16 + 30 + 48 = 148 sq.units
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