1. Warm Up Simplify the fraction: a) 𝟔𝒙 𝟑 b) 𝟑+𝟔𝒙 𝟑 c) 𝟒+𝟑𝒙 𝟓
I will determine the number and types
of roots of a polynomial
Warm Up
Simplify the fraction:
a) 𝟔𝒙 𝟑 b) 𝟑+𝟔𝒙 𝟑 c) 𝟒+𝟑𝒙 𝟓
Solve using the quadratic formula
𝟑 𝒙 𝟐 +𝟔𝒙−𝟒=𝟎
𝑥= −𝑏± 𝑏 2 −4𝑎𝑐 2𝑎

2. 3 3 + 6𝑥 3 4 5 + 3𝑥 5 2𝑥 1+2𝑥 4+3𝑥 5 Warm Up Simplify the fraction:
a) 𝟔𝒙 𝟑 b) 𝟑+𝟔𝒙 𝟑 c) 𝟒+𝟑𝒙 𝟓
𝑥 3
1+2𝑥
𝑥 5
4+3𝑥 5 2𝑥

3. Solve using the quadratic formula
𝟑 𝒙 𝟐 +𝟔𝒙−𝟒=𝟎
𝑥= −(6)± (6) 2 −4(3)(−4) 2(3)
𝑥= −6±
𝑥= −
𝑥= −6−9.17 6
𝑥=.52
𝑥=2.53

4. Fundamental Theorem of Algebra
The Fundamental Theorem of Algebra states that: An nth degree polynomial will have n roots. Example: 𝑦= 𝑥 2 +3𝑥+2 is a second degree polynomial, therefore it has 2 roots. Verify this by graphing it on your calculator.

5. Fundamental Theorem of Algebra
State the number of roots for each polynomial. Verify by graphing on your calculator.
𝑦=4𝑥−5
𝑦=− 𝑥 3 −2 𝑥 2 +29𝑥−30
𝑦= 𝑥 4 −2𝑥−2
What's the problem with equation number 3?
1st degree, 1 x-intercept
3rd degree,
3 x-intercepts

6. Imaginary Numbers What is −1 ? That’s a lie. −1 =𝑖
In the 1500s mathematician Rafael Bombelli invented the concept of 𝑖 so that he could work with −1 .

7. Imaginary Numbers
We can rewrite the square root of a negative number using 𝑖.
Example:
−16 = 16 ∗ −1
−16 =4𝑖
Example 2:
−64 =8𝑖

8. Imaginary Numbers If 𝑖= −1 , what is 𝑖 2 ? 𝑖 2 = −1 2 𝑖 2 =−1
𝑖 2 = −1 2
𝑖 2 =−1
Simplify: 8𝑖 10𝑖
8𝑖 10𝑖 =80 𝑖 2
−80
Simplify: 6−2𝑖+8+4𝑖
14+2𝑖

9. Imaginary Numbers Rewrite the expressions below: −36 −81 6𝑖 100
10𝑖 −3𝑖
4𝑖 2 5𝑖
− 𝑖 2
6 3−2𝑖 +4𝑖 6𝑖 9𝑖 10 30
−80𝑖
−45𝑖
18−8𝑖

10. BREAK

11. Complex Roots 𝑥= −𝑏± 𝑏 2 −4𝑎𝑐 2𝑎
Use the quadratic formula to solve: 0= 𝑥 2 −4𝑥+5 𝑥= −(−4)± (−4) 2 −4(1)(5) 2(1) 𝑥= 4± −4 2 𝑥= 4±2𝑖 2 𝑥=2±𝑖
𝑥= −𝑏± 𝑏 2 −4𝑎𝑐 2𝑎

12. Complex Roots
A real number plus or minus an imaginary number is called a complex number. 𝑦= 𝑥 2 −4𝑥+5 Has two complex roots: 2+𝑖 and 2−𝑖 Graph the function on your calculator. How many times does it cross the x-axis? Since 𝑦= 𝑥 2 −4𝑥+5 does not cross the x-axis, it does not have any real roots.

13. 𝟐 𝒓𝒆𝒂𝒍 𝒓𝒐𝒐𝒕𝒔 + 𝟐 𝒄𝒐𝒎𝒑𝒍𝒆𝒙 𝒓𝒐𝒐𝒕𝒔 = 𝟒𝒕𝒉 𝒅𝒆𝒈𝒓𝒆𝒆
Complex Roots
Let’s revisit 𝑦= 𝑥 4 −2𝑥−2
How many times does the function cross the x-axis?
Based on the Fundamental Theorem of Algebra, how many roots should 𝑦= 𝑥 4 −2𝑥−2 have?
𝑦= 𝑥 4 −2𝑥−2 has 2 real roots and 2 imaginary roots
𝟐 𝒓𝒆𝒂𝒍 𝒓𝒐𝒐𝒕𝒔 + 𝟐 𝒄𝒐𝒎𝒑𝒍𝒆𝒙 𝒓𝒐𝒐𝒕𝒔 = 𝟒𝒕𝒉 𝒅𝒆𝒈𝒓𝒆𝒆

14. Complex Roots 𝑦= 𝑥 2 −4 𝑦= 𝑥 3 +2 𝑦=(𝑥−1)(𝑥+3)(𝑥−6) 2 real
Use your calculator and the Fundamental Theorem of Algebra to determine the number of real and imaginary roots for each polynomial.
𝑦= 𝑥 2 −4
𝑦= 𝑥 3 +2
𝑦=(𝑥−1)(𝑥+3)(𝑥−6)
𝑦=−3 𝑥 4 +2 𝑥 2 −6
2 real
1 real, 2 complex
3 real
4 complex

15. Textbook:
9-90, 9-91, 9-105