1. Factortopia By Alex Bellenie

2. What is Factoring? / Factoring is a process where we find what we multiply in order to get a quantity. / Factoring is effectively “undoing” multiplying. You also using the distributive property backwards. / Factoring is a process where we find what we multiply in order to get a quantity. / Factoring is effectively “undoing” multiplying. You also using the distributive property backwards.

3. Why is it important / Factoring is one of the most important parts of algebra, it is used in a large part of algebra and is a building block of math. / Factoring has many applications it is used to solve quadratic equations, such as 9x 2 +6x+0, and is used to simplify rational expressions. / Factoring is one of the most important parts of algebra, it is used in a large part of algebra and is a building block of math. / Factoring has many applications it is used to solve quadratic equations, such as 9x 2 +6x+0, and is used to simplify rational expressions.

4. Examples / For example when you multiply / 5(x+3)(monomial)(binomial) you would distribute the 5 into both terms ins the parenthesis / Pq(2p 2 q+p+1)(monomial)(trinomial) you distribute pq in to the terms and you would get 2p 3 q 2 +p 2 q+pq / (A+B) 2 (binomial)(binomial) you plug the numbers into a base like this A 2 + 2AB + B 2 / (5x+6)(4x+3) you would use FOIL where you would multiply the First, outside, inside, and last terms to get 20x+15x+24x+18 then you would put like terms together and get 20x+39x+18 / For example when you multiply / 5(x+3)(monomial)(binomial) you would distribute the 5 into both terms ins the parenthesis / Pq(2p 2 q+p+1)(monomial)(trinomial) you distribute pq in to the terms and you would get 2p 3 q 2 +p 2 q+pq / (A+B) 2 (binomial)(binomial) you plug the numbers into a base like this A 2 + 2AB + B 2 / (5x+6)(4x+3) you would use FOIL where you would multiply the First, outside, inside, and last terms to get 20x+15x+24x+18 then you would put like terms together and get 20x+39x+18

5. Examples / For a binomial multiplied by a trinomial like this (2x+3)(4x+4x-2) you would use the box method because Foil will not work / The box method requires one set of terms to be written vertically and the other set to be horizontal you would then make a chart and multiply and combine like terms / and for trinomial multiplied by a trinomial like (4x 2 +3x-5)(7y 2 -5y-2) you would use the box method for this type of multiplication problem / For a binomial multiplied by a trinomial like this (2x+3)(4x+4x-2) you would use the box method because Foil will not work / The box method requires one set of terms to be written vertically and the other set to be horizontal you would then make a chart and multiply and combine like terms / and for trinomial multiplied by a trinomial like (4x 2 +3x-5)(7y 2 -5y-2) you would use the box method for this type of multiplication problem

6. The Factoring Cross / The factoring cross is used to factor problems like these / Ax 2 +bx+c A x C goes in the top / B goes in the bottom / The factoring cross is used to factor problems like these / Ax 2 +bx+c A x C goes in the top / B goes in the bottom

7. The Factoring Cross / To use the cross you find two numbers that when multiplied equal AxC and when added equal B / Ex. X 2 +5x+6 6 2 3 5(x+2)(x+3) / To use the cross you find two numbers that when multiplied equal AxC and when added equal B / Ex. X 2 +5x+6 6 2 3 5(x+2)(x+3)

8. Common Factors / Common factors are easily found in polynomials / They simplify the factoring of Polynomials / First find a number that you can take out of both terms and remove it / 5x+15 can be simplified to 5(x+3) / It is the distributive property used backwards / Try these: 2A 2 +6A+4 4y 2 +8y+16 / Common factors are easily found in polynomials / They simplify the factoring of Polynomials / First find a number that you can take out of both terms and remove it / 5x+15 can be simplified to 5(x+3) / It is the distributive property used backwards / Try these: 2A 2 +6A+4 4y 2 +8y+16

9. Difference of squares / When you multiply: (A+B)(A-B)= A 2 - B 2 / The answer will always have: / two terms and both will be squares / There will be a minus sign between the terms / This is called The Difference of Squares / When you multiply: (A+B)(A-B)= A 2 - B 2 / The answer will always have: / two terms and both will be squares / There will be a minus sign between the terms / This is called The Difference of Squares

10. Difference of squares / Are these a difference of squares? / X 2 -25-36-X 2 4x 2 -25 / To Factor a difference of squares use backwards multiplication / A 2 -B 2 = (A+B)(A-B) / As always with some problems you will be able to factor out a common term / 5-20Y 9 = 5(1-4y 6 )= 5(1+2y 3 ) / Are these a difference of squares? / X 2 -25-36-X 2 4x 2 -25 / To Factor a difference of squares use backwards multiplication / A 2 -B 2 = (A+B)(A-B) / As always with some problems you will be able to factor out a common term / 5-20Y 9 = 5(1-4y 6 )= 5(1+2y 3 )

11. Trinomial Squares / You will get a trinomial square when you multiply (a+b) 2 =a 2 +2ab+b 2 / (a-b) 2 =a 2 -2ab+b 2 / You can determine whether the answer you got is a trinomial square by looking for the following: two of the terms must be squares a 2 and b 2, there is no minus before a 2 and b 2, and the middle term must be +2ab or -2ab / Are these Trinomial Squares? / X 2 +6x+9X 2 +6X+11 / You will get a trinomial square when you multiply (a+b) 2 =a 2 +2ab+b 2 / (a-b) 2 =a 2 -2ab+b 2 / You can determine whether the answer you got is a trinomial square by looking for the following: two of the terms must be squares a 2 and b 2, there is no minus before a 2 and b 2, and the middle term must be +2ab or -2ab / Are these Trinomial Squares? / X 2 +6x+9X 2 +6X+11

12. Factoring X 2 +bx+c / Foil is used for multiplying terms like (x+3)(x+6) / But to Factor their product you simply use Foil in reverse and use the factoring cross / Ax 2 +bx+c / X 2 +7x+10 10 2 5 7 (x+5)(x+2) Try These: X 2 +7x+12X 2 +13x+36 / Foil is used for multiplying terms like (x+3)(x+6) / But to Factor their product you simply use Foil in reverse and use the factoring cross / Ax 2 +bx+c / X 2 +7x+10 10 2 5 7 (x+5)(x+2) Try These: X 2 +7x+12X 2 +13x+36

13. Factoring ax 2 +bx+c Part 2 / This process involves both the cross method and the factoring box / If the leading coefficient is not 1, the product of a will go in the first spot in both (_x+_)(_x+_) and the product of C will go in the second spot in both / This process involves both the cross method and the factoring box / If the leading coefficient is not 1, the product of a will go in the first spot in both (_x+_)(_x+_) and the product of C will go in the second spot in both

14. Factoring ax 2 +bx+c Part 2 / 3x 2 +5x+2 6 2 3 5 3x 2 +2x+3x+2 now you can factor by grouping or use the factoring box X(3x+2)+1(3x+2) (x+1)(3x+2) Try these:6x 2 +7x+28x 2 +10x-3 / 3x 2 +5x+2 6 2 3 5 3x 2 +2x+3x+2 now you can factor by grouping or use the factoring box X(3x+2)+1(3x+2) (x+1)(3x+2) Try these:6x 2 +7x+28x 2 +10x-3

15. Factoring by grouping / X 3 +X 2 +2x+2 next you will add parenthesis but you can add them when you write the problem / (X 3 +x 2 )+(2x+) next you take out a number to make both sets of terms the same / X 2 (X+1)+2(X+1) / Now you put the terms that you took out into a set and put the same terms into one / (X 2 +2)(x+1) / Try these: (8x 3 +2x 2 )+(12x+3) (x 3 +x 2 )+(x+1) / X 3 +X 2 +2x+2 next you will add parenthesis but you can add them when you write the problem / (X 3 +x 2 )+(2x+) next you take out a number to make both sets of terms the same / X 2 (X+1)+2(X+1) / Now you put the terms that you took out into a set and put the same terms into one / (X 2 +2)(x+1) / Try these: (8x 3 +2x 2 )+(12x+3) (x 3 +x 2 )+(x+1)

16. Factoring completely / To find out whether you have factored completely you check many things to find out or “Look” / Look: for a common factor / Look: at the number of terms / Two terms: Difference of squares? / Three terms: Square or Binomial? If not. Test the factor of the terms / Look: to see if you are done…factor completely / To find out whether you have factored completely you check many things to find out or “Look” / Look: for a common factor / Look: at the number of terms / Two terms: Difference of squares? / Three terms: Square or Binomial? If not. Test the factor of the terms / Look: to see if you are done…factor completely

17. Conclusion / Factoring is a highly important process in Algebra and must never ever be overlooked. It is used to solve many tricky problems and is a simple process used to simplify polynomials and many other Algebra terms.