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Factortopia. By Alex Bellenie. 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. Why is it important.

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Factortopia

Factortopia

By Alex Bellenie


What is factoring
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.


Why is it important
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 9x2+6x+0, and is used to simplify rational expressions.


Examples
Examples

  • For example when you multiply

  • 5(x+3)(monomial)(binomial) you would distribute the 5 into both terms ins the parenthesis

  • Pq(2p2q+p+1)(monomial)(trinomial) you distribute pq in to the terms and you would get 2p3q2+p2q+pq

  • (A+B)2 (binomial)(binomial) you plug the numbers into a base like this A2 + 2AB + B2

  • (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


Examples1
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 (4x2+3x-5)(7y2-5y-2) you would use the box method for this type of multiplication problem


The factoring cross
The Factoring Cross

  • The factoring cross is used to factor problems like these

  • Ax2+bx+c A x C goes in the top

  • B goes in the bottom


The factoring cross1
The Factoring Cross

  • To use the cross you find two numbers that when multiplied equal AxC and when added equal B

  • Ex. X2+5x+6

    6

    2 3

    5 (x+2)(x+3)


Common factors
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: 2A2+6A+4 4y2+8y+16


Difference of squares
Difference of squares

  • When you multiply: (A+B)(A-B)= A2-B2

  • 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


Difference of squares1
Difference of squares

  • Are these a difference of squares?

  • X2-25 -36-X2 4x2-25

  • To Factor a difference of squares use backwards multiplication

  • A2-B2= (A+B)(A-B)

  • As always with some problems you will be able to factor out a common term

  • 5-20Y9 = 5(1-4y6)= 5(1+2y3)


Trinomial squares
Trinomial Squares

  • You will get a trinomial square when you multiply (a+b)2=a2+2ab+b2

  • (a-b)2=a2-2ab+b2

  • You can determine whether the answer you got is a trinomial square by looking for the following: two of the terms must be squares a2 and b2, there is no minus before a2 and b2, and the middle term must be +2ab or -2ab

  • Are these Trinomial Squares?

  • X2+6x+9 X2+6X+11


Factoring x 2 bx c
Factoring X2+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

  • Ax2+bx+c

  • X2+7x+10

    10

    2 5

    7

    (x+5)(x+2)

    Try These: X2+7x+12 X2+13x+36


Factoring ax 2 bx c part 2
Factoring ax2+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


Factoring ax 2 bx c part 21
Factoring ax2+bx+c Part 2

  • 3x2+5x+2

    6

    2 3

    5

    3x2+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:6x2+7x+2 8x2+10x-3


Factoring by grouping
Factoring by grouping

  • X3+X2+2x+2 next you will add parenthesis but you can add them when you write the problem

  • (X3+x2)+(2x+) next you take out a number to make both sets of terms the same

  • X2(X+1)+2(X+1)

  • Now you put the terms that you took out into a set and put the same terms into one

  • (X2+2)(x+1)

  • Try these: (8x3+2x2)+(12x+3) (x3+x2)+(x+1)


Factoring completely
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


Conclusion
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.


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