Polynomials. Unit 10 Polynomials 2 of 2 SMART Board Notes.notebook. May 15, 2013

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1 Oct 19 9:41 M errick played basketball for 5 out of the 10 days for four hours each. How many hours did errick spend playing basketball? Oct 19 9:41 M Polynomials Polynomials 1

2 Table of ontents Factors and GF Factoring out GF's Factoring Trinomials x 2 + bx + c Factoring Using Special Patterns Factoring Trinomials ax 2 + bx + c Factoring 4 Term Polynomials Mixed Factoring Solving Equations by Factoring Table of ontents Lesson 8 Factors and Greatest ommon Factors Return to Table of ontents Factors and GF Factors of 10 Factors of 15 Number ank Factors Unique to 10 Factors 10 and 15 have in common Factors Unique to What is the greatest common factor (GF) of 10 and 15? Factors and GF 2

3 Factors of 12 Factors of 18 Factors Unique to 12 Factors 12 and 18 have in common Factors Unique to 18 Number ank What is the greatest common factor (GF) of 12 and 18? Factors and GF 1 What is the GF of 12 and 15? Example Factors and GF 2 What is the GF of 24 and 48? Practice Factors and GF 3

4 3 What is the GF of 28, 56 and 42? Example Factors and GF 4 What is the GF of 14, 28, and 70? Example Factors and GF Variables also have a GF. Rule: The GF of variables is the variable(s) that is in each term raised to the lowest exponent given. Example: Find the GF and Example and and Practic and and and Example Pract Factors and GF 4

5 5 What is the GF of and? Factors and GF 6 What is the GF of and? Factors and GF 7 What is the GF of and? and Example Factors and GF 5

6 8 What is the GF of and and? Practice Factors and GF Lesson 7 #1 6 Lesson 8 #1 6 Oct 16 10:18 M Multiply the following: Oct 19 9:41 M 6

7 Homework Questions: Oct 4 2:08 PM Lesson 9 Factoring out GFs Return to Table of ontents Factoring GF The first step in factoring is to determine its greatest monomial factor. If there is a greatest monomial factor other than 1, use the distributive property to rewrite the given polynomial as the product of this greatest monomial factor and a polynomial. Example 1 Factor each polynomial. a) 6x 4 15x 3 + 3x 2 Find the GF 3x 2 6x 4 15x 3 3x 2 GF: 3x 2 3x 2 3x 2 3x 2 Reduce each term of the polynomial dividing by the GF 3x 2 (2x 2 5x + 1) Factoring GF 7

8 FTORING STEPS 1. Identify the GF 2. ivide each term by GF 3, Rewrite as a product of GF(remaining polynomial) Example 6x 4 15x 3 + 3x 2 1. Find the GF 2. ividing by the GF GF: 3x 2 3x 2 6x 4 15x 3 3x 2 3x 2 3x 2 3x 2 3. Rewrite 3x 2 (2x 2 5x + 1) Factoring GF 4m 3 n 7m 2 n 2 Practice Find the GF GF: m 2 n ivide by the GF Rewrite m 2 n(4n 7n) Factoring GF Sometimes the distributive property can be used to factor a polynomial that is not in simplest form but has a common binomial factor. Example 2 Factor each polynomial. a) y(y 3) + 7(y 3) Find the GF GF: y 3 Reduce each term of the polynomial dividing by the GF (y 3) (y 3)(y + 7) ( y(y 3) (y 3) + 7(y 3) (y 3) ( Factoring GF 8

b) Find the GF GF: Reduce each term of the polynomial dividing by the GF Factoring GF pr 17 11:34 M In

9 Sometimes the distributive property can be used to factor a polynomial that is not in simplest form but has a common binomial factor. Example 2 Factor each polynomial. b) Find the GF GF: Reduce each term of the polynomial dividing by the GF Factoring GF pr 17 11:34 M In working with common binomial factors, look for factors that are opposites of each other. For example: (x y) = (y x) because x y = x + ( y) = y + x = (y x) Factoring GF 9

10 9 True or False: y 7 = 1( 7 + y) True False Factoring GF 10 True or False: 8 d = 1( d + 8) True False Factoring GF 11 True or False: 8c h = 1( 8c + h) True False Factoring GF 10

11 12 True or False: a b = 1( a + b) True False Factoring GF In working with common binomial factors, look for factors that are opposites of each other. For example: (x y) = (y x) because x y = x + ( y) = y + x = (y x) Example 3 Factor each polynomial. a) n(n 3) 7(3 n) Find the GF GF: Reduce each term of the polynomial dividing by the GF (n 3)(n + 7) Factoring GF In working with common binomial factors, look for factors that are opposites of each other. For example: (x y) = (y x) because x y = x + ( y) = y + x = (y x) Example 3 Factor each polynomial. b) p(h 1) + 4(1 h) Find the GF GF: Reduce each term of the polynomial dividing by the GF (h 1)(p 4) Factoring GF 11

12 13 If possible, Factor lready Simplified Factoring GF 14 If possible, Factor lready Simplified Factoring GF 15 If possible, Factor lready Simplified Factoring GF 12

13 16 If possible, Factor lready Simplified Factoring GF 17 If possible, Factor lready Simplified Factoring GF Oct 16 10:18 M 13

14 18 If possible, Factor lready Simplified Factoring GF Please multiply: (x+2)(x+3) Oct 19 9:41 M Please multiply: (x 2)(x 3) Oct 19 9:41 M 14

15 Homework Questions: Oct 4 2:08 PM Homework Questions: Oct 4 2:08 PM Lesson 10 Factoring Trinomials: x 2 + bx + c Return to Table of ontents 15

16 polynomial that can be simplified to the form ax + bx + c, where a 0, is called a quadratic polynomial. Quadratic term. Linear term. onstant term. quadratic polynomial in which b 0 and c 0 is called a quadratic trinomial. If only b=0 or c=0 it is called a quadratic binomial. If both b=0 and c=0 it is a quadratic monomial. Examples: hoose all of the description that apply. ubic Quadratic Linear onstant Trinomial inomial Monomial pr 25 4:47 M 19 hoose all of the descriptions that apply to: E F Quadratic Linear onstant Trinomial inomial Monomial 16

17 20 hoose all of the descriptions that apply to: E F Quadratic Linear onstant Trinomial inomial Monomial 21 hoose all of the descriptions that apply to: E F Quadratic Linear onstant Trinomial inomial Monomial 22 hoose all of the descriptions that apply to: E F Quadratic Linear onstant Trinomial inomial Monomial 17

18 Simplify. 1) (x + 2)(x + 3) = 2) (x 4)(x 1) = 3) (x + 1)(x 5) = 4) (x + 6)(x 2) = nswer ank x 2 5x + 4 x 2 4x 5 x 2 + 5x + 6 x 2 + 4x 12 Slide each polynomial from the circle to the correct expression. RELL What did we do?? Look for a pattern!! To Factor a Trinomial with a Lead oefficient of 1 Recognize the pattern: Factors of 6 have the same signs. Factors of 6 Sum to 5? 1, 6 7 Factors of 6 add to +5. oth factors must be positive. 2, 3 5 pr 25 4:56 M To Factor a Trinomial with a Lead oefficient of 1 Recognize the pattern: Factors of 6 have the same signs. Factors of 6 Sum to 7? Factors of 6 add to 7. oth factors must be negative. 1, 6 7 2, 3 5 pr 25 4:56 M 18

19 Examples: (x 8)(x 1) pr 25 5:18 M 23 The factors of 12 will have what kind of signs given the following equation? oth positive oth Negative igger factor positive, the other negative The bigger factor negative, the other positive 24 The factors of 12 will have what kind of signs given the following equation? oth positive oth negative igger factor positive, the other negative The bigger factor negative, the other positive 19

20 25 Factor (x + 12)(x + 1) (x + 6)(x + 2) (x + 4)(x + 3) (x 12)(x 1) E (x 6)(x 1) F (x 4)(x 3) 26 Factor (x + 12)(x + 1) (x + 6)(x + 2) (x + 4)(x + 3) (x 12)(x 1) E (x 6)(x 1) F (x 4)(x 3) 27 Factor (x + 12)(x + 1) (x + 6)(x + 2) (x + 4)(x + 3) (x 12)(x 1) E (x 6)(x 1) F (x 4)(x 3) 20

21 28 Factor (x + 12)(x + 1) (x + 6)(x + 2) (x + 4)(x + 3) (x 12)(x 1) E (x 6)(x 1) F (x 4)(x 3) To Factor a Trinomial with a Lead oefficient of 1 Recognize the pattern: Factors of 6 have the opposite signs. Factors of 6 Sum to 5? 1, 6 5 2, 3 1 Factors of 6 add to 5. Larger factor must be negative. To Factor a Trinomial with a Lead oefficient of 1 Recognize the pattern: Factors of 6 have the opposite signs. Factors of 6 Sum to 1? 1, 6 5 2, 3 1 Factors of 6 add to +1. Larger factor must be positive. 21

22 Examples pr 19 11:43 M 29 The factors of 12 will have what kind of signs given the following equation? oth positive oth negative igger factor positive, the other negative The bigger factor negative, the other positive 22

23 30 The factors of 12 will have what kind of signs given the following equation? oth positive oth negative igger factor positive, the other negative The bigger factor negative, the other positive 31 Factor x 2 4x 12 (x + 12)(x 1) (x + 6)(x 2) (x + 4)(x 3) (x 12)(x + 1) E (x 6)(x + 2) F (x 4)(x + 3) 32 Factor (x + 12)(x 1) (x + 6)(x 2) (x + 4)(x 3) (x 12)(x + 1) E (x 6)(x + 1) F (x 4)(x + 3) 23

24 33 Factor (x + 12)(x 1) (x + 6)(x 2) (x + 4)(x 3) (x 12)(x + 1) E (x 6)(x + 1) F (x 4)(x + 3) 34 Factor (x + 12)(x 1) (x + 6)(x 2) (x + 4)(x 3) (x 12)(x + 1) E (x 6)(x + 1) F (x 4)(x + 3) Mixed Practice 24

25 35 Factor the following (x 2)(x 4) (x + 2)(x + 4) (x 2)(x +4) (x + 2)(x 4) 36 Factor the following (x 3)(x 5) (x + 3)(x + 5) (x 3)(x +5) (x + 3)(x 5) 37 Factor the following (x 3)(x 4) (x + 3)(x + 4) (x +2)(x +6) (x + 1)(x+12) 25

38 Factor the following (x 2)(x 5) (x + 2)(x + 5) (x 2)(x +5) (x + 2)(x

out. 2) Need 2 numbers that multiply to the constant 3) and add to the

There is no common monomial,so STEP factor: 1 STEP 2 STEP 3 STEP 4

26 38 Factor the following (x 2)(x 5) (x + 2)(x + 5) (x 2)(x +5) (x + 2)(x 5) Steps for Factoring a Trinomial 1) See if a monomial can be factored out. 2) Need 2 numbers that multiply to the constant 3) and add to the middle number. 4) Write out the factors. There is no common monomial,so STEP factor: 1 STEP 2 STEP 3 STEP 4 Steps for Factoring a Trinomial 1) See if a monomial can be factored out. 2) Need 2 numbers that multiply to the constant 3) and add to the middle number. 4) Write out the factors. There is no common monomial,so STEP factor: 1 STEP 2 STEP 3 STEP 4 26

27 Factor: Factor out STEP 1 STEP 2 STEP 3 STEP 4 Factor: Factor out STEP 1 STEP 2 STEP 3 STEP 4 39 Factor completely: 27

28 40 Factor completely: 41 Factor completely: 42 Factor completely: 28

Square of Sums ifference of Sums Product of a Sum and a ifference If we learn to

29 43 Factor completely: Factoring Using Special Patterns Return to Table of ontents Factoring Spec Patterns When we were multiplying polynomials we had special patterns. Square of Sums ifference of Sums Product of a Sum and a ifference If we learn to recognize these squares and products we can use them to help us factor. Factoring Spec Patterns 29

30 Perfect Square Trinomials The Square of a Sum and the Square of a difference have products that are called Perfect Square Trinomials. How to Recognize a Perfect Square Trinomial: Recall: Observe the trinomial The first term is a perfect square. The second term is 2 times square root of the first term times the square root of the third. The sign is plus/minus. The third term is a perfect square. Factoring Spec Patterns Examples of Perfect Square Trinomials Factoring Spec Patterns Is the trinomial a perfect square? rag the Perfect Square Trinomials into the ox. Only Perfect Square Trinomials will remain visible. Factoring Spec Patterns 30

of the first term sign of the middle term sq rt of the third term) 2 Examples: Factoring

31 Factoring Perfect Square Trinomials. Once a Perfect Square Trinomial has been identified, it factors following the form: (sq rt of the first term sign of the middle term sq rt of the third term) 2 Examples: Factoring Spec Patterns 44 Factor Not a perfect Square Trinomial Factoring Spec Patterns 45 Factor Not a perfect Square Trinomial Factoring Spec Patterns 31

46 Factor Not a perfect Square Trinomial Factoring Spec Patterns ifference of Squares The Product of a Sum and a ifference is a difference of Squares.

32 46 Factor Not a perfect Square Trinomial Factoring Spec Patterns ifference of Squares The Product of a Sum and a ifference is a difference of Squares. ifference of Squares is recognizable by seeing each term in the binomial are perfect squares and the operation is subtraction. Factoring Spec Patterns Examples of ifference of Squares Factoring Spec Patterns 32

33 Is the binomial a difference of squares? rag the ifference of Squares binomials into the ox. Only ifference of Squares will remain visible. Factoring Spec Patterns Factoring a ifference of Squares Once a binomial is determined to be a ifference of Squares, it factors following the pattern: (sq rt of 1 st term sq rt of 2 nd term)(sq rt of 1 st term + sq rt of 2 nd term) Examples: Factoring Spec Patterns 47 Factor Not a ifference of Squares Factoring Spec Patterns 33

Factoring Spec Patterns 50 Factor using ifference of

34 48 Factor Not a ifference of Squares Factoring Spec Patterns 49 Factor Not a ifference of Squares Factoring Spec Patterns 50 Factor using ifference of Squares: Not a ifference of Squares Factoring Spec Patterns 34

51 Factor Factoring Spec Patterns Factoring Trinomials: ax 2 + bx + c Return to Table of ontents Factoring Trinomials a isn't 1 How to factor a trinomial of the form ax² + bx + c.

35 51 Factor Factoring Spec Patterns Factoring Trinomials: ax 2 + bx + c Return to Table of ontents Factoring Trinomials a isn't 1 How to factor a trinomial of the form ax² + bx + c. Example: Factor 2d² + 15d + 18 Find the product of a and c : 2 18 = 36 Now find two integers whose product is 36 and whose sum is equal to b or 15. Factors of 36 Sum = 15? 1, 36 2, 18 3, = = = 15 Now substitute into the equation for 15. 2d² + (12 + 3)d + 18 istribute 2d² + 12d + 3d + 18 Group and factor GF 2d(d + 6) + 3(d + 6) Factor common binomial (d + 6)(2d + 3) Remember to check using FOIL! Factoring Trinomials a isn't 1 35

= 13? 1, 30 2, 15 3, 10 5, 6 1 + 30 = 31 2 + 15 = 17 3 + 10 = 13 5 + 6 = 11 Factoring Trinomials a isn't 1 2 2b b 10 a = 2, c = 10, and b = 1 Since a times c is negative,

36 Factor. 15x² 13x + 2 a = 15 and c = 2, but b = 13 Since both a and c are positive, and b is negative we need to find two negative factors of 30 that add up to 13 Factors of 30 Sum = 13? 1, 30 2, 15 3, 10 5, = = = = 11 Factoring Trinomials a isn't 1 Factor. 2 2b b 10 a = 2, c = 10, and b = 1 Since a times c is negative, and b is negative we need to find two factors with opposite signs whose product is 20 and that add up to 1. Since the sum is negative, larger factor of 20 must be negative. Factors of 20 Sum = 1? 1, 20 2, 10 4, = = = 1 Factoring Trinomials a isn't 1 Factor 6y² 13y 5 Factoring Trinomials a isn't 1 36

37 polynomial that cannot be written as a product of two polynomials is called a prime polynomial. Factoring Trinomials a isn't 1 52 Factor Prime Polynomial Factoring Trinomials a isn't 1 53 Factor Prime Polynomial Factoring Trinomials a isn't 1 37

54 Factor Prime Polynomial Factoring Trinomials a isn't 1 Factoring 4 Term Polynomials Return to Table of ontents Factoring 4 terms Polynomials with four terms like ab 4b + 6a 24, can be factored by

38 54 Factor Prime Polynomial Factoring Trinomials a isn't 1 Factoring 4 Term Polynomials Return to Table of ontents Factoring 4 terms Polynomials with four terms like ab 4b + 6a 24, can be factored by grouping terms of the polynomials. Example 1: ab 4b + 6a 24 (ab 4b) + (6a 24) Group terms into binomials that can be factored using the distributive proper b(a 4) + 6(a 4) Factor the GF (a 4) (b + 6) Notice that a 4 is a common binomial factor and factor! Factoring 4 terms 38

39 Example 2: 6xy + 8x 21y 28 (6xy + 8x) + ( 21y 28) Group 2x(3y + 4) + ( 7)(3y + 4) Factor GF (3y +4) (2x 7) Factor common binomial Factoring 4 terms You must be able to recognize additive inverses!!! (3 a and a 3 are additive inverses because their sum is equal to zero.) Remember 3 a = 1(a 3). Example 3: 15x 3xy + 4y 20 (15x 3xy) + (4y 20) Group 3x(5 y) + 4(y 5) Factor GF 3x( 1)( 5 + y) + 4(y 5) Notice additive inverses 3x(y 5) + 4(y 5) Simplify (y 5) ( 3x + 4) Factor common binomial Remember to check each problem by using FOIL. Factoring 4 terms 55 Factor 15ab 3a + 10b 2 (5b 1)(3a + 2) (5b + 1)(3a + 2) (5b 1)(3a 2) (5b + 1)(3a 1) Factoring 4 terms 39

40 56 Factor 10m 2 n 25mn + 6m 15 (2m 5)(5mn 3) (2m 5)(5mn+3) (2m+5)(5mn 3) (2m+5)(5mn+3) Factoring 4 terms 57 Factor 20ab 35b a (4a 7)(5b 9) (4a 7)(5b + 9) (4a + 7)(5b 9) (4a + 7)(5b + 9) Factoring 4 terms 58 Factor a 2 ab + 7b 7a (a b)(a 7) (a b)(a + 7) (a + b)(a 7) (a + b)(a + 7) Factoring 4 terms 40

Mixed Factoring Return to Table of ontents Mixed Factoring Summary of Factoring Factor the Polynomial 2 Terms ifference of Squares Perfect Square Trinomial Factor out GF 3 Terms Factor the Trinomial

41 Mixed Factoring Return to Table of ontents Mixed Factoring Summary of Factoring Factor the Polynomial 2 Terms ifference of Squares Perfect Square Trinomial Factor out GF 3 Terms Factor the Trinomial 4 Terms Group and Factor out GF. Look for a ommon inomial a = 1 a = 1 heck each factor to see if it can be factored again. If a polynomial cannot be factored, then it is called prime. Mixed Factoring Examples 3r 3 9r 2 + 6r 3r(r 2 3r + 2) 3r(r 1)(r 2) Mixed Factoring 41

42 59 Factor completely: Mixed Factoring 60 Factor completely prime polynomial Mixed Factoring 61 Factor prime polynomial Mixed Factoring 42

43 62 Factor completly prime polynomial Mixed Factoring 63 Factor Prime Polynomial Mixed Factoring Solving Equations by Factoring Return to Table of ontents Solving Equations by Factoring 43

44 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 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 = 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? How would you solve it? We can use the Zero Product Property to solve it. How can we turn this polynomial into a multiplication problem? Factor it! Factoring yields: (x 6)(x + 4) = 0 y the Zero Product Property: x 6 = 0 or x + 4 = 0 fter solving each equation, we arrive at our solution: { 4, 6} Solving Equations by Factoring 44

Solve Solving Equations by Factoring Zero

than zero subtract to make one side of the

The teacher tells the class that the height

When the rocket hits the ground, its height

45 Solve Solving Equations by Factoring Zero Product rule works only when the product of factors equals zero. If the equation equals some value other than zero subtract to make one side of the equation zero. Example Solving Equations by Factoring science class launches a toy rocket. The teacher tells the class that the height of the rocket at any given time is h = 16t t. When will the rocket hit the ground? When the rocket hits the ground, its height is 0. So h=0 which can be substituted into the equation: The rocket had to hit the ground some time after launching. The rocket hits the ground in 20 seconds. The 0 is an extraneous (extra) answer. Solving Equations by Factoring 45

46 64 hoose all of the solutions to: E F Solving Equations by Factoring 65 hoose all of the solutions to: E F Solving Equations by Factoring 66 hoose all of the solutions to: E F Solving Equations by Factoring 46

47 67 ball is thrown with its height at any time given by When does the ball hit the ground? 1 seconds 0 seconds 9 seconds 10 seconds Solving Equations by Factoring Jul 6 7:09 PM 47

Polynomials. Factors and Greatest Common Factors. Slide 1 / 128. Slide 2 / 128. Slide 3 / 128. Table of Contents

Polynomials. Factors and Greatest Common Factors. Slide 1 / 128. Slide 2 / 128. Slide 3 / 128. Table of Contents Slide 1 / 128 Polynomials Table of ontents Slide 2 / 128 Factors and GF Factoring out GF's Factoring Trinomials x 2 + bx + c Factoring Using Special Patterns Factoring Trinomials ax 2 + bx + c Factoring