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

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 3 3 + 6𝑥 3 1+2𝑥 4 5 + 3𝑥 5 4+3𝑥 5 2𝑥

Solve using the quadratic formula 𝟑 𝒙 𝟐 +𝟔𝒙−𝟒=𝟎 𝑥= −(6)± (6) 2 −4(3)(−4) 2(3) 𝑥= −6± 84 6 𝑥= −6+9.17 6 𝑥= −6−9.17 6 𝑥=.52 𝑥=2.53

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.

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

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 .

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

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

Imaginary Numbers Rewrite the expressions below: −36 −81 6𝑖 100 10𝑖 −3𝑖 4𝑖 2 5𝑖 −25 3𝑖 2 6 3−2𝑖 +4𝑖 6𝑖 9𝑖 10 30 −80𝑖 −45𝑖 18−8𝑖

BREAK

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

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.

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

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

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