Quadratic Relationships

Unit #2 - Factored Form

What you will learn:

1. Expanding and simplifying
2. Monomial Factoring (GCF)
3. Binomial Factoring (GCF)
4. Factoring by grouping (4 terms)
5. Simple trinomial factoring
6. Complex trinomial factoring
7. Special Product- Difference of Squares
8. Special Product- Perfect Square Trinomial
9. Word Problem using factored form

10. Video of Factoring

About Factored Form

Factor form: y= a (x-s)(x-r)



  1. a = direction of opening/ stretched or compressed
  • if a = >0 then it opens upward and has a min. value
  • if a = <0 then it opens downward and has a max. value
  • if a = <0 it is compressed (wide)
  • if a = >0 is stretched (narrow)



  • (x-s) and (x-r)= the x-intercepts (zeros, roots) of the parabola, this is where the parabola intersects on either side of the x axis




  • the x intercept is the opposite sign of the factor
    ex. (x-2) -------> x = (2,0)
  • This is due to the zero product rule: if two numbers multiply to equal zero, one or both of the numbers must equal zero.
ex. a x b = 0 so a or b is 0 (or both)


Finding x intercepts


Step 1: Set both factors to 0

Step 2: Use opposite sign when it comes out of bracket


0 = a (x-2)(x-4)

0 = (x-2) |||| 0=(x-4)

X= 2 ||| x= 4

X= (2,0) ||| x= (4,0)


Finding AOS (axis of symmetry)


Step 1: Add both x values together

Step 2: Divide the sum by two


AOS = 2+4

---------

2

= 6

-----

2

= 3


Finding Vertex [coordinate y]:


Step 1: Plug in AOS into original factored equation to solve for y


(x-2)(x-4)

= (3-2) (3-4)

= (1)(-1)

y = -1

Vertex: (3,-1)


--------------------------


Finding the y-intercept


Set x= 0


Ex.

y= -4(x-2) (x-4)

y=-4(0-2) (0-4)

y= -4 (-2)(-4)

y= -32


Y intercept = (0,-32)


Determining a:

Step 1: sub in vertex (x,y) into equation and solve for a


Vertex: (3,2)

4 = a (3-2) (3-4)

4 = a (1) (-1)

4 = a (-1)

----------

-1

-4 = a


y= -4 (x-2)(x-4)

Big image

1. Expanding and simplifying

  • Step 1: Multiply each of the terms in the first binomial with each of the terms in the second binomial
    Step 2: Collect like terms

    Tips:
  • Utilize BEDMAS at all times
  • Remember exponent rules!

Ex.

(x+2) (2x+3)
= 2x² + 3x + 4x + 6

= 2x² +7x +6

2. Monomial Factoring

Step 1: Find the GCF ( can be coefficients and variables)
Step 2: Divide each term by GCF

  • GCF comes outside bracket
  • What ever is left (not common) goes inside the bracket


Ex.

5c + 10d

= 5c + 10d

-------------------

5


= 5 ( c+2d)


Tips:

  • Not all questions can be solved. If nothing is common it is not possible.
  • Ex. 3c + 5f

= not possible

  • When exponents are involved, use the lowest common exponent as the GCF

3. Binomial Factoring

Step 1: Look for two binomials that are exactly the same, that serves as a common factor.

Step 2: What is left, is the also a factor.

Ex.

5x (3x+2) +4x (3x+2)
= (3x+2) (5x+4)

4. Factor by Grouping

Step 1: Group the terms into two by adding brackets around the terms that share a common factor.
Step 2: Factor the grouped terms
Step 3: You will notice that two of the factors are the same, this is one of the factors.

Step 4: What is left (outside of the bracket) will make up the other factor


Ex.
9x² + 15x + 3x + 5

= (9x² + 15x) (+3x + 5)

-----------------

3x

= 3x (3x+5) + 1 (3x+5)

= (3x+5) (3x+1)


Tips:

  • Pay attention to whether the sign before the second bracket is positive/negative. Change signs! (distribute the sign)

ex. - (4x - 3)

= (-4x + 3)

  • If nothing is common, then 1 is

ex. (+3x + 5)

= 1 (3x + 5)

  • If the answer produces two of the same common factors you can write it once in a bracket squared

ex. 16m² - 12m -12m + 9

= (4m -3) (4m - 3)
= (4m-3) ²

5. Simple Trinomial Factoring

Formula for simple trinomial: ax² + bx+ c

(no coefficient in front of a )


  • x = variable
  • a , b, c = constants
  • a = 1 (not zero)


Standard form ---------> factored form

x² + bx + c = (x+r) (x+s)


  • b = r+s
  • c = rs
  • r and s are integers


Step 1: Find two numbers whose product is c and whose sum is b

ex. x² + 12x + 27

__ x ___ = 27 ------------> 9 x 3 = 27

__ + ___ = 12 -------------> 9 + 3 = 27


x² + 12x + 27

= (x 9) (x 3)


Step 2: Look at the signs of b and c:

  • +b and +c = both r and s are positive
  • -b and +c = both r and s are negative
  • -c = one of r or s is negative
  • -b and -c = one of r or s is negative


x² + 12x + 27
( b and c are both positive)
= (x+9) (x+3)

6. Complex Trinomial Factoring

Formula for complex trinomial : ax² + bx + c (a does not equal 1)


Step 1: Find two numbers which product is ac and sum is b.

Step 2: Break up the middle term (bx),by writing the two terms which sum is b [Decomposition]

Step 3: Factor by grouping (place brackets around terms with common factors)


Ex. 3x² + 8x + 4

ac = 3 x 4
= 12
___ x ____ = 12 --------------------> 6x2 = 12
___ + ____ = 8 ----------------------> 6+2 = 12


3x² + 8x + 4
= 3x² + 6x + 2x + 4

= (3x² =6x) + (2x + 4)
---------------- --------------
3x 2
= 3x (x +2) + 2 (x+2)

= (x+2) (3x+2)


Tips:

  • Always look for GCF first!

Ex. 16x² + 26x - 12

= -------------------------

2

= 2 (8x² + 13x - 6)

= 2 (8x² + 16x - 3x - 6)

= 2 (8x² + 16x) - (3x - 6)

= 2 ------------------ ----------

8x 3

= 2 [(8x (x+2) - 3 (x+2)]

= 2 (x + 2) (8x - 3)

7. Special Product - Difference of Squares

Formula for difference of squares: a² - b²


What is difference of squares?

  • When you multiply the sum and difference of two terms and the two middle terms are opposite (cancel out), so they add to zero


Expand and Simplify:

Ex. (g+6) (g-6)

= g² -6g + 6g - 36 [middle terms cancel]

= g² - 36


Factoring:


Step 1: Identify two perfect squares

Step 2: Write in format (x + ) (x - )

Step 3: Plug in square root of value b


Ex.

x² - 49

= (x+ ) (x - )
= √49 = 7

= (x + 9) (x - 9)

Tips:

  • ALWAYS one of the numbers is negative one is positive (opposite)

8. Special Product - Perfect Square Trinomials

Formula: ax² + 2ab + b²


What is perfect square trinomials?

  • When you multiply a binomial by itself the product is always a perfect square trinomial (PST)



Expanding and Simplifying:

Step 1: Square the first term

Step 2: Twice the product of the terms = middle term

Step 3: Collect like terms

Step 4: Square the last term

Ex.

(a+b) ² = a² + 2ab +b²

or

(a-b) = a² - 2ab + b²


(x+2)² = (x+2) (x+2)

x² + 2(x)(2) + 2

x² + 2x + 2x + 4

x² + 4x + 2²


Factoring:

x² + 4x + 2²

= x² + (2) 4x/2 + 2²

= x² + 2x + 2x + 4

= (x+2x) + (2x+4)

= ----------- ----------

x 2

= x (x+2) 2 (x+ 2)

= (x + 2) (x +2)

= (x+2)²


Tips:

  • b cannot be negative!
  • if there is nothing in front of the x when finding GCF, 1 is common

9. Word Problem using factored form

A rectangle has a given area: 8x² + 2x - 15


a) Factor to find the algebraic expression for the length and width of the rectangle


a = 8x² + 12x - 10x - 15

= (8x² + 12x) - (10x - 15)

= ----------------- -------------

4x (2x+3) -5 (2x+3)

a= (2x+3) (4x-5)


b) If x= 12, determine the perimeter and area of the rectangle

Step 1: Plug in x values to previous answer


a= (2x+3) (4x-5)

a = [2(12)+3] [4(12)-5]

a = (24+3) (48-5)

a = (27) (43)

a = 1161 cm²


-------------------------------------------------


p = 2l + 2w

p = 2 (2x+3) + 2 (4x-5)

p = 4x + 6 + 8x - 10

p = 12x - 4

p = 12(12) - 4

p = 144 - 4

p = 140 cm


Tips:

  • when finding GCF, the sign is taken from the larger number
  • for area remember to add unit ²
  • for perimeter remember to add the unit

10. Factoring Video

❤² How to Solve Quadratic Equations By Factoring (mathbff)

Fun Factor Song!

Teach Me How To Factor (WSHS Math Rap Song)