Special Products and Factors.pptx

Special Products and different techniques in factoring

1. 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. Special rules are d eveloped so that products may be found mentally. The process of finding a Short method is called SPECIAL PRODUCTS .

5. Some special products are: Product of Two Binomials Square of Binomials Cube of Binomial Product of the sum and difference

6. Product of two binomials Examples : (2x – 3)(x – 7)

7. The product of two binomials with like terms is equal to the product of the first terms of the binomials plus the sum of the products of the two outer terms and two inner terms plus the product of the last terms of the binomials (FOIL Method).

8. Multiply the following: 1. (2x+3) (x+8) 2. (4x+2) (3x-5) 3. (5x-2) (3x+7)

9. Square of a binomial Examples : (2x – 3y) 2

10. The square of a binomial is equal to the square of the first term plus twice the product of the two terms plus the square of the second term.

11. Activity: 1. (X+6) 2 2. (3x+9) 2 3. (4y- 7) 2 4. (2x-8y) 2 5. (5x+6) 2

12. Product of sum and difference Examples : (2x+3y 2 )(2x – 3y 2 )

13. The product of the sum and difference of two terms is equal to the square of the first term minus the square of the second term.

14. Activity: 1. (3x+y)(3x-y) 2. (4x+5)(4x-5) 3. (7y+8)(7y-8) 4. (3x+5)(3x-5) 5. (5x-8)(5x+8)

15. Cube of a binomial Examples : (x + y 2 ) 3 (2x – 3y) 3

16. The cube of a binomial is equal to the cube of the first term plus thrice the product of the second term and the square of the first term plus thrice the product of the first term and the square of the second term plus the cube of the second term.

17. Activity 1. (3x+4) 3 2. (2x+3) 3 3. (3x+5) 3

18. Product of a binomial and a trinomial Examples : (x + y 2 )(x 2 – xy 2 + y 4 ) (2x – 3y)(4x 2 + 6xy + 9y 2 ) (x – y)(x 2 + 2xy + y 2 )

19. The product of a binomial (x + y) and a trinomial (x 2 – xy + y 2 ) is simply equal to the sum of the cubes of the first term and second term of the binomial.

20. Factoring A polynomial with integer coefficients is said to be prime if it has no monomial or polynomial factors with integer coefficients other than itself and one . A polynomial with integer coefficients is said to be factored completely when each of its polynomial factor is prime .

21. The reverse of finding the related products is a process called factoring . Rewriting a polynomials as a product of polynomial factors is call Factoring Polynomials. These are: Common Monomial Factor Difference of two Squares Factoring Trinomials Factoring Perfect Square Trinomial Factoring by Grouping

22. Factoring a common monomial factor Examples : 4x 2 yz 3 – 2x 4 y 3 z 2 +6 x 3 y 2 z 4

23. Difference of two squares Examples : 4u 2 – 9v 2 (a+2b) 2 – (3b + c) 2

24. Factoring trinomials Examples : x 2 + 7x + 10 a 2 – 10a + 24 a 2 + 4ab – 21b 2 20x 2 + 43xy + 14y 2

25. Perfect square trinomials Examples : y 2 – 10y + 25 24a 3 + 72a 2 b + 54ab 2

26. Sum and difference of cubes Examples : 27 – x 3 t 3 + 8

27. 3y 6 – 6y 3 + 3

28. Factoring by GROUPING Examples: 10a 3 + 25a – 4a 2 – 10 Common factor

29. Factoring by GROUPING Examples: 3xy – yz + 3xw – zw

30. 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.

31. Solve for the following special products. 1. (3a + 2b) (3a + 2b) 6. (3x – 2y)3 2. (a – 8) (a – 6) 7. (8x – 5 + 2y)2 3. (2x + 5)3 8. (4c2 + 2d) (4c2 – d) 4. (12k3 – 6) (12k3 + 6) 9. (g3 + h2) 5. (7x + y) (7x – y) 10. (2x + 3y) (2x - 3y)

32. Factor the following 1. 4x 2 -9 2. 100-81x 2 3. x 2 – 7x + 12 4. 2x 2 – xy – 55y 2 5. 2xy + 6y + x +3 6. 9x 2 + 6xy + y 2 7. 8m 3 n 3 – 125

33. Reflection When do we use special products? Enumerate the special products we discussed in this unit. When is a polynomial completely factored? Enumerate the different types of factoring.