THE RING OF POLYNOMIALS. Special Products and Factoring

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1 THE RING OF POLYNOMIALS Special Products and Factoring

2 Special Products and Factoring Upon completion, you should be able to Find special products Factor a polynomial completely

3 Special Products - rules for finding products in a faster way - long multiplication is the last resort

4 Product of two binomials ax b cx d acx ad bc x bd ( ax by)( cx dy) acx ( ad bc) xy bdy Examples: 1. x5x6 x x x3x7 3x5x3 x 17x 1 6x x 15

5 Square of a binomial A B A AB B Examples: 1. x y x 4xy 4y. 3. x 3y 3w 4v 3 4x 1xy 9y w 4w v 16v

6 Product of Sum and Difference A B A B A B Examples: 1. 3x 5y3x 5y 9x 5y. x 3y x 3y 4x 9y 4

7 Cube of a Binomial A B A 3A B 3AB B Examples: 1.. x y 3 x 3y 3 x 3x y 3xy y x 36x y 54xy 7y 3 3

8 Product of a Binomial and a Trinomial 3 3 A B A AB B A B Examples: 4 1. x y x xy y x 3 y 6. x 3y 4x 6xy 9y 8x 7y 3 3

9 Square of a Trinomial A B C A B C AB AC BC Examples: 1. x y w 4 x y w xy xw y w. x 3y z 4x 9y z 1xy 4xz 6yz

10 Factoring - process of writing an expression as a product of its factors - reverse process of finding products

11 Factoring Example: Factor Example: Factor 70 completely For numbers, factoring completely is the same as finding the prime decomposition of a number.

12 Factoring A polynomial is factorable in R if it can be expressed as a product of two or more polynomials with rational coefici ents.

13 Factoring A polynomial with rational coefficients is said to be factored completely when each of its polynomial factor is prime.

14 Factoring a Common ab ac a b c Monomial Factor Examples: 1. 8x 4x 4x x x 3x x 3 x x x 3 1

15 Difference of Two Squares A B A B A B 105 Examples: y. 4u 9v 3. a b 3b c 4 y4 y u 3vu 3v a b 3b c a b 3b c (06) 84

16 Example Factor x 4 4 y completely. 4 4 x y x y x y x y x y x yx y

17 Example x y x yi x yi x y x y x y But these violate the restrictions hence, 4 4 x y x y x y x y is a complete factorization.

18 acx Examples: 1. x 7x 10. a 10a 4 3.a 4ab 1b Factoring Trinomials acx ad bc x bd ax b cx d ( ad bc) xy bdy 4. 0x 43xy 14y ( ax by)( cx x 5x a 6a 4 a 7ba 3b 4x 7y5x y dy)

19 Perfect Square Trinomials A AB B A B Examples: 1. y 10y 5. 16x 8x x 30xy 5y a 7a b 54ab y 5 4x 1 3x5y 6a 4a 1ab 9b 6a a 3b

20 Sum and Difference of Cubes 3 3 A B A B A AB B Examples: 1. 7 x 3.t y y x x y y 3 x 9 3x x t t t 4 3 y 1 x 3 y 3 y 1 y y 1 x y x xy y

21 Factoring by Grouping Examples: a 5a 4a 10 5a a 5 a 5 5a a 5. 3xy yz 3xw zw a 7b a 3b 3 3 y x z w x z y w3x z a 3b a 3ab 9b a 3b a 3b a 3ab 9b 1

22 Factoring by Grouping Examples: ) ( 1 1 z w z w z w z w z w z wz w z wz w

23 Factoring by Grouping Examples: a a b b? Volunteer?

24 Special Products and Factoring Summary Factors ax bcx d A B A B A B A B A AB B A B 3 A B C Products acx ad bc x bd A AB B A A B 3 3 B 3 3 A 3A B 3AB B A B C AB AC BC

25 Special Products and Factoring Summary Always look for a common monomial factor, FIRST. Factoring can also be done by grouping some terms to yield a common polynomial factor.

26 With rational coefficient x y x y x xy y 3 3 x y y x y xy y x x

27 Special Products and Factoring Exercises Find the following products as fast as you can: 1.[( x 3 y) 5][3( x 3 y) ].[ w ( y z)] ( x a y a )( x a y a ) 4.(3x 5 y ) 3

28 Special Products and Factoring Exercises Find the following products as fast as you can: 4.(w 3 xy ) (3x 5 y )(9x 15xy 5 y ) 4 6.(x 3 y )(4x 6xy 9 y ) 7.( x xy 3 y ) 4

29 Special Products and Factoring Exercises Factor completely: x y z x y z x yz. 5a b 36c m 1 4. ( y1) 7( y1) 3

30 Special Products and Factoring Exercises Factor completely: w 3 6. ( x 1) 7 7. ( x z) ( z x) x 3x x 3

31 Special Products and Factoring Exercises Factor completely: 3 9. x 9x x x y 4y x xy y x y ax b bx a 13. x xy y 7x 7y 10

32 Reflection 1. When do we use special products?. Enumerate the special products we discussed in this unit. 3. When is a polynomial completely factored? 4. Enumerate the different types of factoring. We factor polynomials to simply expressions, and ultimately for easy solving of equations

33 Properties of the Set of Polynomials with + and 1. The sum of two polynomials is a polynomial.. Addition of polynomials is associative. 3. Is there an additive identity? 4. Is there an additive inverse for each polynomial?

34 Properties of the Set of Polynomials with + and 5. Addition of polynomials is Commutative. The set of polynomials together with + is an abelian group.

35 Properties of the Set of Polynomials with + and 6. The product of two poynomials is a polynomial. 7. Multiplication of polynomials is Associative. 8. Multiplication is distributive over addition of polynomials.

36 Properties of the Set of Polynomials with + and The set of polynomials together with + and is a ring. Is it a commutative ring? What is the multiplicative identity? Is it a field?