Factoring and Box Method
- 1. Algebra 2 Factoring Basics & Box Method
- 2. Factoring Polynomials This process is basically the REVERSE of the distributive property. distributive property ( x + 2)( x − 5) = x − 3 x − 10 factoring 2
- 3. Factoring Polynomials In factoring you start with a polynomial (2 or more terms) and you want to rewrite it as a product (or as a single term) Three terms x − 3 x − 10 = ( x + 2)( x − 5) 2 One term
- 4. Techniques of Factoring Polynomials 1. Greatest Common Factor (GCF). The GCF for a polynomial is the largest monomial that divides each term of the polynomial. Factor out the GCF: 4y − 2y 3 2
- 5. Factoring Polynomials - GCF 4y − 2y 3 2 Write the two terms in the form of prime factors… 22y y y 2 y y 2 yy ( 2 y They have in common 2yy 1) = 2 y (2 y − 1) 2 This process is basically the reverse of the distributive property.
- 6. Check the work…. 2 y (2 y − 1) = 4 y − 2 y 2 3 2
- 7. Factoring Polynomials - GCF 3 terms Factor the GCF: 4ab − 12a b c + 8ab c = 3 2 3 2 4 a b ( b - 3a c 2 One term 4 2 + 2b c 2 2 )
- 8. Factoring Polynomials - GCF EXAMPLE: 5 x(2 x − 4) − 3(2 x − 4) = (2 x − 4) ( 5x - 3 )
- 9. Examples Factor the following polynomial. 12 x − 20 x = 3 ⋅ 4 ⋅ x ⋅ x − 4 ⋅ 5 ⋅ x ⋅ x ⋅ x ⋅ x 2 4 = 4 ⋅ x ⋅ x (3 − 5 ⋅ x ⋅ x ) = 4 x (3 − 5 x ) 2 2
- 10. Examples Factor the following polynomial. 15 x y + 3 x y = 3 ⋅ 5 ⋅ x ⋅ y + 3 ⋅ x ⋅ y 3 5 2 4 3 5 = 3 ⋅ x ⋅ y (5 ⋅ x ⋅ y + 1) 2 4 = 3 x 2 y 4 (5 xy + 1) 2 4
- 11. Techniques of Factoring Polynomials 2. Factoring a Polynomial with four or more Terms by Grouping x + 3x + 2 x + 6 = There is no GCF for all four terms. x ( x + 3) + 2 ( x + 3) = In this problem we factor GCF by grouping the first two terms and the last two terms. 3 2 2 ( x + 3) ( x + 2) 2
- 13. Techniques of Factoring Polynomials 3. Factoring Trinomials. x + 5x + 6 2 We need to find factors of 6 ….that add up to 5 Since 6 can be written as the product of 2 and 3 and 2 + 3 = 5, we can use the numbers 2 and 3 to factor the trinomial.
- 14. Factoring Trinomials, continued... x + 5x + 6 2 2x3=6 2+3=5 Use the numbers 2 and 3 to factor the trinomial… Write the parenthesis, with An “x” in front of each. (x Write in the two numbers we found above. ( x + 2 )( x + 3 ) )( x )
- 15. Factoring Trinomials, continued... So we factored the trinomial… x + 5 x + 6 = ( x + 2 )( x + 3 ) 2 You can check your work by multiplying back to get the original answer ( x + 2 )( x + 3 ) = x + 3 x + 2 x + 6 = 2 = x + 5x + 6 2
- 16. Factoring Trinomials x + 7x + 6 2 Find factors of 6 that add up to 7 6 and 1 x − 5x − 6 2 Find factors of – 6 that add up to –5 – 6 and 1 x + 1x − 6 2 Find factors of – 6 that add up to 1 3 and –2
- 17. Factoring Trinomials x + 7x + 6 2 ( x + 6 )( x + 1 ) factors of 6 that add up to 7: x − 5x − 6 2 6 x + 1x − 6 1 ( x − 6 )( x + 1 factors of – 6 that add up to – 5: – 6 2 and ) and 1 ( x + 3 )( x − 2 ) factors of – 6 that add up to 1: 3 and – 2
- 18. Factoring Trinomials The hard case – “Box Method” 2x + x − 6 2 Note: The coefficient of x2 is different from 1. In this case it is 2 2 2 x +1x − 6 First: Multiply 2 and –6: 2 (– 6) = – 12 Next: Find factors of – 12 that add up to 1 – 3 and 4
- 19. Factoring Trinomials The hard case – “Box Method” 2x + x − 6 2 1. Draw a 2 by 2 grid. 2. Write the first term in the upper left-hand corner 3. Write the last term in the lower right-hand corner. 2x 2 −6
- 20. Factoring Trinomials The hard case – “Box Method” 2x + x − 6 2 Find factors of – 12 that add up to 1 – 3 x 4 = – 12 –3+4=1 1. Take the two numbers –3 and 4, and put them, complete with signs and variables, in the diagonal corners, like this: 2 2x 4x –3 x −6 It does not matter which way you do the diagonal entries!
- 21. The hard case – “Box Method” 1. Then factor like this: Factor Top Row x 2 2x 4x − 3x −6 From Left Column 2x 2 x 2x 2 4x − 3x −6 Factor Bottom Row x 2 2 2x 4x − 3x −6 From Right Column 2x 2 x 2x 2 4x −3 − 3x −6
- 22. The hard case – “Box Method” −3 − 3x −6 2x 2 x 2x +2 +4 x Note: The signs for the bottom row entry and the right column entry come from the closest term that you are factoring from. DO NOT FORGET THE SIGNS!! Now that we have factored our box we can read off our answer: 2 x + x − 6 = ( x + 2)(2 x − 3) 2
- 23. The hard case – “Box Method” 4 x − 19 x + 12 = 2 Look for factors of 48 that add up to –19 x 2 4 x 4x 3 − 3x – 16 and – 3 4 − 16 x 12 4 x − 19 x + 12 = ( 4 x − 3)( x − 4) 2 Finally, you can check your work by multiplying back to get the original answer.
- 24. Use “Box” method to factor the following trinomials. 1. 2x2 + 7x + 3 2. 4x2 – 8x – 21 3. 2x2 – x – 6
- 25. Check your answers. 1. 2x2 + 7x + 3 = (2x + 1)(x + 3) 2. 2x2 – x – 6 = (2x + 3)(x – 2) 3. 4x2 – 8x – 21 = (2x – 7)(2x + 3)
- 26. Note… Not every quadratic expression can be factored into two factors. • For example x2 – 7x + 13. We may easily see that there are no factors of 13 that added up give us –7 • x2 – 7x + 13 is a prime trinomial.
- 27. Factoring the Difference of Two Squares a2– ab + ab – b2 = a2 – b2 (a + b)(a – b) = FORMULA: a2 – b2 = (a + b)(a – b) The difference of two bases being squared, factors as the product of the sum and difference of the bases that are being squared.
- 28. Factoring the difference of two squares a2 – b2 = (a + b)(a – b) Factor x2 – 4y2 Difference of two squares (x) 2 Factor 16r2 – 25 2 (2y) (x – 2y)(x + 2y) Now you can check the results… Difference Of two squares 2 (4r) 2 (5) (4r – 5)(4r + 5)
- 29. Difference of two squares y − 16 = 2 = ( y ) − (4) 2 2 = ( y − 4)( y − 4)
- 30. Difference of two squares 25 x − 81 = 2 = (5 x ) − (9) 2 2 = (5 x − 9)(5 x + 9)
- 31. Difference of two squares y − 16 = 4 = ( y ) − ( 4) 2 2 2 = ( y − 4)( y + 4) 2 2 = ( y − 2)( y + 2)( y + 4) 2