1. Warm Up Simplify: a) 𝟓 𝟐+𝟑𝒊 −𝟔𝒊 b) 𝟐𝒊 𝟒𝒊 𝟔𝒊
I can completely factor a polynomial
Warm Up
Simplify:
a) 𝟓 𝟐+𝟑𝒊 −𝟔𝒊 b) 𝟐𝒊 𝟒𝒊 𝟔𝒊
c) (𝟒−𝟑𝒊)(𝟓−𝒊) d) −𝟐(𝟕+𝟐𝒊)(𝟑−𝒊)
Divide
𝟒 𝒙 𝟑 −𝟒 𝒙 𝟐 +𝟗𝒙−𝟏𝟐 𝟐𝒙+𝟏

2. Simplify:
a) 𝟓 𝟐+𝟑𝒊 −𝟔𝒊 b) 𝟐𝒊 𝟒𝒊 𝟔𝒊
𝟏𝟎+𝟏𝟓𝒊−𝟔𝒊
𝟏𝟎+𝟗𝒊
𝟖 𝒊 𝟐 𝟔𝒊
−𝟖 𝟔𝒊
−𝟒𝟖𝒊

3. Simplify:
c) (𝟒−𝟑𝒊)(𝟓−𝒊) d) −𝟐(𝟕+𝟐𝒊)(𝟑−𝒊)
𝟐𝟎−𝟒𝒊−𝟏𝟓𝒊+𝟑 𝒊 𝟐
𝟐𝟎−𝟏𝟗𝒊+𝟑 −𝟏
𝟏𝟕−𝟏𝟗𝒊
(−𝟏𝟒−𝟒𝒊)(𝟑−𝒊)
−𝟒𝟐+𝟏𝟒𝒊−𝟏𝟐𝒊+𝟒 𝒊 𝟐
−𝟒𝟐+𝟐𝒊+𝟒 −𝟏
−𝟒𝟔+𝟐𝒊

4. Divide
𝟒 𝒙 𝟑 −𝟒 𝒙 𝟐 +𝟗𝒙−𝟏𝟐 𝟐𝒙+𝟏 2𝑥
4 𝑥 3
−6𝑥 2
12𝑥 +1
2𝑥 2
−3𝑥 6
2𝑥 2
−3𝑥 6
𝟐 𝒙 𝟐 −𝟑𝒙+𝟔− 𝟏𝟖 𝟐𝒙+𝟏

5. Homework Questions

6. What do you notice about the constant?
Multiply
𝑥+1 𝑥−7
𝑥−7 𝑥+3
𝑥+2 8𝑥−12
(𝑥+4) 2
𝑥 2 −6𝑥−7
𝑥 2 −4𝑥−21
8 𝑥 2 +4𝑥−24
𝑥 2 +8𝑥+16
What do you notice about the constant?

7. Identifying Possible Zeros
𝑦= 𝑥 4 − 𝑥 3 −5 𝑥 2 +3𝑥+6
Based on the constant, what are some possible zeros?
Factors of 6:
1∗6 , −1∗−6 , 2∗3 , −2∗−3
𝑥=±1, 𝑥=±6, 𝑥=±2, 𝑥=±3
Graph the equation on your calculator and find one zero

8. Identifying Possible Zeros
𝑦= 𝑥 4 − 𝑥 3 −5 𝑥 2 +3𝑥+6
A zero at 𝑥=2 means a linear factor of:
(𝑥−2)
Divide
𝑥 4 − 𝑥 3 −5 𝑥 2 +3𝑥+6 𝑥−2

9. (𝑥−2)(𝑥 3 + 𝑥 2 −3𝑥−3) 𝑥 𝑥 4 1 𝑥 3 −3𝑥 2 −3𝑥 −2 −2 𝑥 3 −2𝑥 2 6𝑥 6 𝑥 3
𝑥 4 − 𝑥 3 −5 𝑥 2 +3𝑥+6 𝑥−2 𝑥
𝑥 4
1 𝑥 3
−3𝑥 2
−3𝑥 −2
−2 𝑥 3
−2𝑥 2 6𝑥 6
𝑥 3
𝑥 2
−3𝑥 −3
(𝑥−2)(𝑥 3 + 𝑥 2 −3𝑥−3)

10. (𝑥−2)(𝑥+1)( 𝑥 2 −3) 𝑥 𝑥 3 0 𝑥 2 −3𝑥 +1 𝑥 2 0𝑥 −3 𝑥 2 0𝑥 −3
𝑥 3 +𝑥 2 −3𝑥−3 𝑥+1 𝑥
𝑥 3
0 𝑥 2
−3𝑥 +1
𝑥 2 0𝑥 −3
𝑥 2 0𝑥 −3
(𝑥−2)(𝑥+1)( 𝑥 2 −3)

11. Completely Factor the Polynomial
𝑦= 𝑥 4 −6 𝑥 3 −6 𝑥 2 −6𝑥−7
Identify possible factors by looking at the constant
Find one linear factor (using your calculator) and divide
Divide by a second linear factor
Write polynomial in factored form
Identify all roots (real and complex) of the polynomial

12. Completely Factor the Polynomial
𝑦= 𝑥 4 −6 𝑥 3 −6 𝑥 2 −6𝑥−7
Identify possible factors by looking at the constant
𝑥=±7 or 𝑥=±1

13. Completely Factor the Polynomial
𝑦= 𝑥 4 −6 𝑥 3 −6 𝑥 2 −6𝑥−7
Find one linear factor (using your calculator) and divide
𝑥 4 −6 𝑥 3 −6 𝑥 2 −6𝑥−7 𝑥+1
(𝑥+1)(𝑥 3 −7 𝑥 2 +𝑥−7)

14. Completely Factor the Polynomial
𝑦= 𝑥 4 −6 𝑥 3 −6 𝑥 2 −6𝑥−7
Divide by a second linear factor
𝑥 3 −7 𝑥 2 +𝑥−7 𝑥−7
(𝑥+1)(𝑥−7)( 𝑥 2 +1)

15. Completely Factor the Polynomial
𝑦= 𝑥 4 −6 𝑥 3 −6 𝑥 2 −6𝑥−7
Write polynomial in factored form
Identify all roots (real and complex) of the polynomial
(𝑥+1)(𝑥−7)( 𝑥 2 +1)
𝑥=7 , 𝑥=−1and 𝑥=±𝑖

16. A few reminders…
Don’t forget to factor out a greatest common factor, when possible.
Example:
3 𝑥 4 +6 𝑥 3 −3 𝑥 2 +9𝑥
3𝑥 𝑥 3 +2 𝑥 2 −𝑥+3
When you get to a quadratic, you can use the quadratic formula to find the roots if it doesn’t factor

17. Factoring and Finding Roots WS