EQ: How can all of the roots of a polynomial (both real & imaginary) be found?

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Presentation transcript:

EQ: How can all of the roots of a polynomial (both real & imaginary) be found?

1. x = a is a root 2. P(a) = 0 3. x – a is a factor of the polynomial 4. Dividing by x – 4 gives a remainder of 0

 Every polynomial of degree “n” has exactly “n” roots and “n” linear factors. 2x 3 + 5x 2 – x + 10 Degree:____ #of factors:_____ #of roots: _____

 Find all real zeros of  Set the function equal to 0  Factor out GCF and factor completely  Set the factors equal to zero & solve Factored Form: _______________________________ Roots: _______________________________________

 Find all the roots & factors of f(x) = x 3 – 64  Set the function equal to 0  Factor out GCF & factor completely  Set the factors equal to zero → Solve/use quadratic formula Factored Form: _______________________________ Roots: _______________________________________

 #1 Find all the roots & factors of f(x) = x Factored Form: _______________________________ Roots: _______________________________________

 #2 Find all the roots & factors of f(x) = 8x 3 – 125 Factored Form: _______________________________ Roots: _______________________________________

 Step 1: Find the real roots by graphing  Step 2: Synthetically Divide using one of the roots  Step 3: Quadratic → Quadratic Formula NOT → Synthetically divide with another root  Step 4: Repeat until you have a quadratic  Step 5: Use Quadratic Form

1: Find the real roots by graphing 2: Synthetically Divide using one root 3: Synthetically divide with other root 4: Repeat until you have a quadratic 5: Use Quadratic Form

 #1 Degree: _______ Total number of zeros: ________ Real zero at x = _______ How many imaginary must there be? ________ Find the remaining zeros:

 #2 Degree: _______ Total number of zeros: ________ Real zero at x = _______ How many imaginary must there be? ________ Find the remaining zeros:

 Close friends Stacey, Una, and Amir were all born on July 4 th. Stacey is one year older than Una. Una is two years younger than Amir. On July 4, 2010, the product of their ages was 2300 more than the sum of their ages. How old is each friend on that day?

f(x) = x 5 - 4x x x x + 36 Degree:_______ Total number of factors: ________ Degree:_______ Total number of factors: ________

 Use synthetic division & the quadratic formula to factor: