# 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)**

- 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.

**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)

## 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 + b**²

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