New Jersey enter 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. NJTL 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. lick to go to website: www.njctl.org www.njctl.org lgebra I Polynomials Part 1 2013 03 01 www.njctl.org www.njctl.org Table of ontents efinitions of Monomials, Polynomials and egrees dding and Subtracting Polynomials Multiplying Monomials ividing by Monomials Mulitplying a Polynomial by a Monomial Multiplying Polynomials Special inomial Products Solving Equations by Factoring Table of ontents 1
Welcome ack!!!! Hope everyone had a great Spring reak!! Let's get our brains working again... I took a plane from Philadelphia, P to Tamp FL. It took approximately 2 hours and 10 minutes and covered about 927 miles. How fast was the plane flying? 1) Solve my problem. 2) Write your own math problem about something you did over Spring reak. e reative and write it on a piece of paper to hand in. I will use them as For Starters problems. Lesson 1 Lesson 1 efinitions of Monomials, Polynomials and egrees Return to Table of ontents Lesson 1 efinition monomial: one term expression formed by a number, a variable, or the product of numbers and variables. Examples of monomials... 81y 4 z 17x 2 28 32,457 rt 6 4x mn 3 ef of Monomial 2
rag the following terms into the correct sorting box. If you sort correctly, the term will be visible. If you sort incorrectly, the term will disappear. Monomials xy 4 7 12 a + b 5 5x + 7 15 6+5rs 7x 3 y 5 4 x 2 (5 + 7y) 48x 2 yz 3 4(5a 2 bc 2 )t 16 Monomial efinition polynomial: n expression that contains two or more monomials. Examples of polynomials... 5+a 2 8x 3 +x 2 7+b+c 2 +4d 3 rt 6 + c 2 +d a 4 b 15 8a 3 2b 2 4c mn 3 ef of Polynomial egrees of Monomials efinition degree of a monomial: The sum of the exponents of its variables. *The degree of any nonzero number such as 5 or 12 is 0. *The number 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. constant has a degree 0, because there is no variable. egree 3
Practice: egree Practice: egree Practice: egree 4
Practice: egree egrees 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. egree of a Poly Find the degree of each polynomial nswers: 1) 3 1) 0 2) 12c 3 2) 3 3) ab 4) 8s 4 t 5) 2 7n 6) h 4 8t 7) s 3 + 2v 2 y 2 1 3) 2 4) 5 5) 1 6) 4 7) 4 example example practice practice example egree of a Poly 5
Example egree of a Poly Example egree of a Poly Practice egree of a Poly 6
Practice egree of a Poly Lesson 10.1 page 3 #1 16 Oct 16 10:18 M Homework: 1) 3 2) 7 3) 7 4) 8 5) 12 6) 2 7) 6 8) 8 9) 5 10) 3 11) 4 12) 11 13) 12 14) 2 15) 8 16) 14 Lesson 2 7
Lesson 2 dding and Subtracting Polynomials Return to Table of ontents Lesson 2 dd/subt Poly Standard Form The terms of the polynomial must be 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: dd/subt Poly Review... Monomials with the same variables and the same power are like terms. Like Terms Unlike Terms 4x and 12x 3b and 3a x y and 4x y 3 3 6a b and 2ab 2 2 dd/subt Poly 8
Review... ombine these like terms using the indicated operation. Remember... add or subtract coefficients, do not change variables. dd/subt Poly 1 Simplify dd/subt Poly 2 Simplify dd/subt Poly 9
3 Simplify dd/subt Poly To add polynomials, combine the like terms from each poly 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 = dd/subt Poly 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 dd/subt Poly 10
4 dd Example dd/subt Poly 5 dd Example dd/subt Poly 6 dd Practice dd/subt Poly 11
7 dd Practice dd/subt Poly 8 dd Example dd/subt Poly To subtract polynomials, subtract the coefficients of like terms. Example: 3x 4x = 7x 13y ( 9y) = 22y 6xy 13xy = 7xy dd/subt Poly 12
9 Subtract Example dd/subt Poly 10 Subtract Practice dd/subt Poly 11 Subtract Example dd/subt Poly 13
12 Subtract Practice dd/subt Poly 13 Subtract Practice dd/subt Poly 14 What is the perimeter of the following figure? (answers are in units) dd/subt Poly 14
Lesson 10.2 Page 4 #1 18 Oct 16 10:18 M 10.2 Homework: 1) 40d 2 2) 56f 6 3) 8e 5 4) 42t 32 5) 48x 6 y 5 6) 72x 8 y 8 7) 5a 2 b 7 c 8) 27m 12 n 12 9) x 32 y 12 10) e 24 f 12 11) m 56 n 16 12) 81y 16 z 14 13) 64r 12 s 24 t 3 14) 8h 36 j 24 15) 20,736x 26 y 22 16) 8,192a 44 b Lesson 3 Lessons 3&4 Review Multiplying Monomials ividing Monomials Return to Table of ontents Lesson 3 15
Recall: a monomial is a one term expression formed by a number, a variable, or the product of numbers and variables. Now that we reviewed how to add and subtract polynomials, we need to review how to multiply them. Let's start with two monomials. x 7 an you remember the rule that can be used as a short cut for finding the product of powers? Rule: a m a n = a m + n Mult Monomials Multiplying monomials with coefficients. (6 7) (x 3 x 4 ) 6x 3 7x 4 42x 7 Mult Monomials Multiplying Monomials 3x 7 y 3 9x 4 y 6 ( 3 9) (x 7 x 4 ) (y 3 y 6 ) 27x 11 y 9 Mult Monomials 16
Now, remember this rule? Slide to check. Mult Monomials Power of a Power Rule: Examples 1) Slide to check. 2) Slide to check. Mult Monomials ividing Monomials Return to Table of ontents ividing Monomials 17
onsider this: x 7 x 3 Write this in expanded notation. x x x x x x x Simplify. x x x 1 1 1 x x x x x x x = x 4 x x x 1 1 1 an you devise a rule that can be used as a short cut for dividing monomials with like bases? ividing Monomials Examples ividing Monomials with Like ases Rule: ividing Monomials x Now, consider this: 3 x 5 x x x Write this in expanded notation. x x x x Simplify. 1 1 1 x x x 1 = x x x x x x 2 1 1 1 If we did this using the short cut of subtracting the exponents, we'd have: x 3 = x 3 5 = x x 5 2 Since it's impossible to get two different answers for this problem, we conclude that: x ividing Monomials 18
Examples: Negative Exponent Rule For all real numbers,. Negative Exponents Zero Exponent Rule For all non zero real numbers,. ividing Monomials Rules Rules 19
Lesson 10.3 # 1 16 Lesson 10.4 # 1 16 Oct 16 10:18 M Mechi had to work monday Sunday over the break. He made $6.50 per hour and worked 8 hours each day. How much money did Mechi make over the break? 10.3 Homework: 1) 40d 2 2) 56f 6 3) 8e 5 4) 42t 32 5) 48x 6 y 5 6) 72x 8 y 8 7) 5a 2 b 7 c 8) 27m 12 n 12 9) x 32 y 12 10) e 24 f 12 11) m 56 n 16 12) 81y 16 z 14 13) 64r 12 s 24 t 3 14) 8h 36 j 24 15) 20,736x 26 y 22 16) 8,192a 44 b 10.4 Homework: 1) f 3 /e 2) h 2 /j 3) r 4 /s 8 4) 6/5n 5) y 3 /x 7 6) 1 7) 1 8) 1 9) u 9 v 10) hj 5 11) 1/r 6 s 3 12) 7m 11 /5 13) x 2 y 2 / 2 14) f 20 /e 10 15) 1 16) 8x 6 Lesson 5 period 1 Jazmin slept for 12 hours every day over the break. How many hours did she sleep in all? 10.3 Homework: 1) 40d 2 2) 56f 6 3) 8e 5 4) 42t 32 5) 48x 6 y 5 6) 72x 8 y 8 7) 5a 2 b 7 c 8) 27m 12 n 12 9) x 32 y 12 10) e 24 f 12 11) m 56 n 16 12) 81y 16 z 14 13) 64r 12 s 24 t 3 14) 8h 36 j 24 15) 20,736x 26 y 22 16) 8,192a 44 b 10.4 Homework: 1) f 3 /e 2) h 2 /j 3) r 4 /s 8 4) 6/5n 5) y 3 /x 7 6) 1 7) 1 8) 1 9) u 9 v 10) hj 5 11) 1/r 6 s 3 12) 7m 11 /5 13) x 4 y 2 / 2 14) f 20 /e 10 15) 1 16) 8x 6 Lesson 5 Period 2 20
Lesson 10.5 Multiplying a Polynomial by a Monomial Return to Table of ontents Poly x Mono Rule: 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) 10x 3 + 12x 2 + 16x 10x 3 + 12x 2 16x Poly x Mono 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 Poly x Mono 21
Let's Try It! Multiply to simplify. 1. 2x 4 + 4x 3 7x Slide to check. 2 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 to check. Poly x Mono 15 What is the area of the rectangle shown? x 2 x 2 + 2x + 4 Poly x Mono 16 6x 2 + 8x 12 6x 2 + 8x 2 12 6x 2 + 8x 2 12x 6x 3 + 8x 2 12x Poly x Mono 22
17 Poly x Mono 18 Poly x Mono 19 Find the area of a triangle (= 1 / 2bh) with a base of 4x and a height of 2x 8. ll answers are in square units. Poly x Mono 23
Lesson 10.5 #1 12 Oct 16 10:18 M Homework Questions: Oct 4 2:08 PM Homework Questions: Oct 4 2:08 PM 24
Homework Questions: Oct 4 2:08 PM Homework Questions: Oct 4 2:08 PM 10.5 Homework: 1) 3d 3 +4d 2) 4x 5 3x 3 3) 8a 3 b 4 4) Mariana worked for 6 hours a day for one week and got paid $7.50 per hour. how much did she earn? 5) 6) 7) 8) 9) 10) 11) 12) Lesson 6 Period 1 25
10.5 Homework: 1) 3d 3 +4d 2) 4x 5 3x 3 3) 8a 3 b 4 4) Frank played video games for 5 hours a day over the break. If he played video games every day, how many hours of his life did he waste over the beak? 5) 6) 7) 8) 9) 10) 11) 12) Lesson 6 Period 1 Lesson 6 Multiplying Polynomials Return to Table of ontents Mult Polynomials Find the total area of the rectangles. 2 5 8 6 rea of the big rectangle rea of the horizontal rectangles rea of each box sq.units Mult Polynomials 26
Let us observe the work from the previous example, sq.units From to, we changed the problem so that instead of a polynomial times a polynomial, we now have a monomial times a polynomial. Use this to help solve the next example. Mult Polynomials Find the total area of the rectangles. x 2x 4 3 Mult Polynomials pr 12 11:15 M 27
pr 12 11:15 M To multiply a polynomial by a polynomial, you multiply each term of the first polynomials by each term of the second. Then, add like terms. Example 1: (2x + 4y)(3x + 2y) 2x(3x + 2y) + 4y(3x + 2y) 2x(3x) + 2x(2y) + 4y(3x) + 4y(2y) 6x 2 + 4xy + 12xy + 8y 2 6x 2 + 16xy + 8y 2 Example 2: Mult Polynomials pr 12 11:20 M 28
pr 12 11:24 M The FOIL Method can be used to remember how multiply two binomials. To multiply two binomials, find the sum of... First terms Outer terms Inner Terms Last Terms Example: First Outer Inner Last Mult Polynomials Try it! Find each product. 1) (x 4)(x 3) x 2 7x Slide + 12 to check. 2) (x + 2)(2x 2 3x 8) 2x Slide to check. 3 + x 2 14x 16 Mult Polynomials 29
Try it! 3) (2x 3y)(4x + 5y) Find each product. 8x 2 2xy 15y 2 Slide to check. 4) (x 2 + 3x 6)(x 2 2x + 4) Slide to check. x 4 + x 3 8x 2 + 24x 24 pr 24 7:52 M pr 12 11:30 M 20 What is the total area of the rectangles shown? 2x 4 4x 5 Mult Polynomials 30
pr 12 11:33 M pr 12 11:38 M 21 Feb 20 8:04 M 31
22 Mult Polynomials 23 Mult Polynomials 24 Mult Polynomials 32
25 Find the area of a square with a side of Mult Polynomials 26 What is the area of the rectangle (in square units)? Feb 28 2:10 PM How would we find the area of the shaded region? Shaded rea = Total area Unshaded rea sq. units pr 24 8:09 M 33
27 What is the area of the shaded region (in sq. units)? 11x 2 + 3x 8 7x 2 + 3x 9 7x 2 3x 10 11x 2 3x 8 Feb 28 2:10 PM 28 What is the area of the shaded region (in square units)? 2x 2 2x 8 2x 2 4x 6 2x 2 10x 8 2x 2 6x 4 Feb 28 2:10 PM Special inomial Products Return to Table of ontents Special inomial Products 34
Square of a Sum (a + b) 2 (a + b)(a + b) a 2 + 2ab + b 2 The square of a + b is the square of a plus twice the product of a and b plus the square of b. Example: (5x + 3) 2 (5x + 3)(5x + 3) 25x 2 + 30x + 9 Special inomial Products Square of a ifference (a b) 2 (a b)(a b) a 2 2ab + b 2 The square of a b is the square of a minus twice the product of a and b plus the square of b. Example: (7x 4) 2 (7x 4)(7x 4) 49x 2 56x + 16 Special inomial Products Product of a Sum and a ifference (a + b)(a b) a 2 + ab + ab + b 2 Notice the ab and ab a 2 b 2 equals 0. The product of a + b and a b is the square of a minus the square of b. Example: (3y 8)(3y + 8) Remember the inner and 9y 2 64 outer terms equals 0. Special inomial Products 35
Try It! Find each product. 1. (3p + 9) 2 9p 2 + Slide 54p to + 81 check. 2. (6 p) 2 36 Slide 12p to + pcheck. 2 3. (2x 3)(2x + 3) 4x 2 Slide 9 to check. Special inomial Products 29 x 2 + 25 x 2 + 10x + 25 x 2 10x + 25 x 2 25 Special inomial Products 30 Special inomial Products 36
31 What is the area of a square with sides 2x + 4? Special inomial Products 32 Special inomial Products hris L. walked to K mart one day over break. It took him 5 mins to walk there and it was one half mile from his house. How fast was hris walking? Homework 10.6 1. 12x 2 +11x+2 2. v 4 4 3. 3h 3 2h 2 15h+10 4. 6r 3 +5r 2 7r 4 5. 12x 3 18x 2 9x+15 6. 2x 2 +7x 10 7. 12f 2 +37f+21 8. 12g 2 18g 12 9. 4r 4 20r 2 +25 10. 6w 3 +7w 2 10w 8 11. 10h 3 32h 2 8h 30 12. 12x 2 14x 11 Oct 19 9:41 M 37
Homework 10.6 1. 12x 2 +11x+2 2. v 4 4 3. 3h 3 2h 2 15h+10 4. 6r 3 +5r 2 7r 4 5. 12x 3 18x 2 9x+15 6. 2x 2 +7x 10 7. 12f 2 +37f+21 8. 12g 2 18g 12 9. 4r 4 20r 2 +25 10. 6w 3 +7w 2 10w 8 11. 10h 3 32h 2 8h 30 12. 12x 2 14x 11 Oct 19 9:41 M Homework Questions: Oct 4 2:08 PM Homework Questions: Oct 4 2:08 PM 38
Homework Questions: Oct 4 2:08 PM Lesson 7 Zero Product Property Return to Table of ontents Lesson 7 Given the following equation, what conclusion(s) can be drawn? ab = 0 Since the product is 0, one of the factors, a or b, must be 0. This is known as the Zero Product Property. Solving Equations by Factoring 39
Zero Product Property Rule: If ab=0, then either a=0 or b=0 pr 24 8:16 M Given the following equation, what conclusion(s) can be drawn? (x 4)(x + 3) = 0 Since the product is 0, one of the factors must be 0. Therefore, either x 4 = 0 or x + 3 = 0. x 4 = 0 or x + 3 = 0 + 4 + 4 3 3 x = 4 or x = 3 Therefore, our solution set is { 3, 4}. To verify the results, substitute each solution back into the original equation. To check x = 3: (x 4)(x + 3) = 0 To check x = 4: (x 4)(x + 3) = 0 ( 3 4)( 3 + 3) = 0 ( 7)(0) = 0 0 = 0 (4 4)(4 + 3) = 0 (0)(7) = 0 0 = 0 Solving Equations by Factoring What if you were given the following equation? (x 6)(x + 4) = 0 Example y the Zero Product Property: x 6 = 0 or x + 4 = 0 x = 6 x = 4 fter solving each equation, we arrive at our solution: { 4, 6} Solving Equations by Factoring 40
33 Solve (a + 3)(a 6) = 0. {3, 6} { 3, 6} { 3, 6} {3, 6} Practice Mar 27 5:22 M 34 Solve (a 2)(a 4) = 0. {2, 4} { 2, 4} { 2, 4} {2, 4} Example Mar 27 5:22 M 35 Solve (2a 8)(a + 1) = 0. { 1, 16} { 1, 16} { 1, 4} { 1, 4} Practice Mar 27 5:22 M 41
Jul 6 6:19 PM 42