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

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

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

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

Homework Questions

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?

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

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

(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)

(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)

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

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

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)

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)

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

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

Factoring and Finding Roots WS