THE RING OF POLYNOMIALS Special Products and Factoring
Special Products and Factoring Upon completion, you should be able to Find special products Factor a polynomial completely
Special Products - rules for finding products in a faster way - long multiplication is the last resort
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 11 30. 3. x3x7 3x5x3 x 17x 1 6x x 15
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 4 3 6 9w 4w v 16v
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
Cube of a Binomial 3 3 3 A B A 3A B 3AB B Examples: 1.. x y 3 x 3y 3 x 3x y 3xy y 3 4 6 8x 36x y 54xy 7y 3 3
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
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
Factoring - process of writing an expression as a product of its factors - reverse process of finding products
Factoring Example: Factor 70. 7 10 980 890 18 40 36 0 4180 Example: Factor 70 completely. 4 3 5 For numbers, factoring completely is the same as finding the prime decomposition of a number.
Factoring A polynomial is factorable in R if it can be expressed as a product of two or more polynomials with rational coefici ents.
Factoring A polynomial with rational coefficients is said to be factored completely when each of its polynomial factor is prime.
Factoring a Common ab ac a b c Monomial Factor Examples: 1. 8x 4x 4x x1 5 4 3. x 3x x 3 x x x 3 1
Difference of Two Squares A B A B A B 105 Examples: 101 1. 16 y. 4u 9v 3. a b 3b c 4 y4 y u 3vu 3v a b 3b c a b 3b c 105101 105101 4(06) 84
Example Factor x 4 4 y completely. 4 4 x y x y x y x y x y x yx y
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.
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)
Perfect Square Trinomials A AB B A B Examples: 1. y 10y 5. 16x 8x 1 3. 9x 30xy 5y 3 4. 4a 7a b 54ab y 5 4x 1 3x5y 6a 4a 1ab 9b 6a a 3b
Sum and Difference of Cubes 3 3 A B A B A AB B Examples: 1. 7 x 3.t 8 3 6 3 3.y y 1 6 3 3 6 4.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
Factoring by Grouping Examples: 3 1. 10a 5a 4a 10 5a a 5 a 5 5a a 5. 3xy yz 3xw zw 3 3 3.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
Factoring by Grouping Examples: 1 1 1 1 1 ) ( 1 1 z w z w z w z w z w z wz w z wz w
Factoring by Grouping Examples: a a b b? Volunteer?
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
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.
With rational coefficient x y x y x xy y 3 3 x 3 3 3 y y x 64 4 3 y xy y x x 4 4 16
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)] 3 3 3.( x a y a )( x a y a ) 4.(3x 5 y ) 3
Special Products and Factoring Exercises Find the following products as fast as you can: 4.(w 3 xy ) 3 4 5.(3x 5 y )(9x 15xy 5 y ) 4 6.(x 3 y )(4x 6xy 9 y ) 7.( x xy 3 y ) 4
Special Products and Factoring Exercises Factor completely: 1. 3 6 3 4 3 3 x y z x y z x yz. 5a b 36c 4 3. 16m 1 4. ( y1) 7( y1) 3
Special Products and Factoring Exercises Factor completely: 6 5. 64 w 3 6. ( x 1) 7 7. ( x z) ( z x) 3 3 3 8. x 3x x 3
Special Products and Factoring Exercises Factor completely: 3 9. x 9x x 18 10. x y 4y 4 11. 3 3 x xy y x y 1. 3 3 ax b bx a 13. x xy y 7x 7y 10
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
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?
Properties of the Set of Polynomials with + and 5. Addition of polynomials is Commutative. The set of polynomials together with + is an abelian group.
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.
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?