M3D11 Have out: Bellwork: x – 4 x2 + 4x + 42 x + 2y x2 – 2xy + (2y)2 Assignment, red pen, highlighter, notebook M3D11 Have out: Bellwork: Factor completely by using either the sum or difference of cubes or group factoring. 1) 2) 3) x – 4 x2 ( )( ) + 4x + 42 ( )( ) x + 2y x2 – 2xy + (2y)2 +1 +1 +2 +1 +2 total: +1 +1

Finding all Zeros of P(x) and Writing Polynomial Equations Part I: Solve for the zeros for each polynomial function. Steps: 1. Set P(x) = ___ Factor 2. ________ completely, if possible. 3. Set each factor equal to ___ by the Zero Product Property. 4. Solve for x. Find all solutions (both ______ and ________). Hint: If the polynomial is degree 2, there are ___ zeros. If degree 3, then ___ zeros. etc. real non–real 2 3

1) P(x) = x4 – 1 Wait a minute! There is a problem with this. x4 – 1 = 0 How many answers do we have? 2 + 1 + 1 x4 = 1 What is the degree? 4 How many zeros should we have? 4

So what happened to the other 2 solutions? 1) P(x) = x4 – 1 Let’s solve it another way: + 1 + 1 x4 – 1 = 0 difference of squares x4 = 1 (x2 + 1)(x2 – 1) = 0 (x2 + 1)(x + 1)(x – 1) = 0 x2 + 1= 0 x + 1 = 0 x – 1 = 0 – 1 –1 –1 –1 +1 +1 x = –1 x = 1 x2 = –1 If you had just solved, you will lose your imaginary solutions. ALWAYS factor completely to help you solve whenever the degree is greater than two.

3) P(x) = x2 + 2x + 5 4) P(x) = x4 – x2 – 20 x2 + 2x + 5 = 0 –5 –5 (x2 + 4)(x2 – 5) = 0 x2 + 2x = –5 + 1 + 1 x2 + 4 = 0 x2 – 5 = 0 x2 + 2x + 1 = –4 –4 –4 +5 +5 (x + 1)2 = –4 x2 = –4 x2 = 5 –1 –1

Work on problems #2, 5, 6 #2  use group factoring Hints: #6  use difference of cubes

2) P(x) = x3 – 4x2 + x – 4 x3 – 4x2 + x – 4 = 0 x2(x – 4) + 1(x – 4) = 0 (x – 4)(x2 + 1) = 0 x – 4 = 0 x2 + 1 = 0 –1 –1 +4 +4 x2 = –1 x = 4

5) P(x) = x3 – 2x2 – 3x + 6 6) P(x) = x3 – 8 x3 – 2x2 – 3x + 6 = 0 +2 +2 x2 + 2x ___ = –4 ___ + 1 + 1 x2 – 3 = 0 x – 2 = 0 +3 +3 +2 +2 x = 2 (x + 1)2 = –3 x2 = 3 x = 2

When finding the zeros of a polynomial function, we notice that complex zeros always are a _________ ______. conjugate pair Part II: Find an equation of a polynomial with the given zeros. Write it in standard form. Steps: 1. Write each zero as a ______ in the polynomial equation. factor 2. Expand the polynomial by using ______ or ________ rectangles. F.O.I.L. generic 3. Check: If there are 3 zeros, then the polynomial is degree ___. Etc. 3

Examples: 1) Find a quadratic P(x) with zeros 3, –2. x = 3 x = –2 Set each zero equal to x. x – 3 = 0 x + 2 = 0 Set them equal to zero. Set equal to P(x) and multiply the factors together using F.O.I.L. or generic rectangles. P(x) = (x – 3)(x + 2) P(x) = x2 – x – 6 Note: the directions state to “Find an equation of a polynomial” so you do not need to find a specific leading coefficient, “a”.

2) Find a cubic P(x) with zeros 1, ±i. Set each zero equal to x. –1 –1 Set them equal to zero. x – 1 = 0 Recall #2. This is how we got ±i. + 1 + 1 x2 + 1 = 0 –1 –1 x2 + 1 = 0 Reverse the process! x2 = –1 P(x) = (x – 1)(x2 + 1) Set the factors equal to P(x) and expand using F.O.I.L. or generic rectangles. P(x) = x3 – x2 + x – 1

Complete exercises #4 – 6. 3) Find a quartic P(x) with zeros ± 3, ±4i. Quartic P(x) means “4th degree” P(x) P(x) = (x2 – 3)(x2 + 16) P(x) = x4 + 16x2 – 3x2 – 48 P(x) = x4 + 13x2 – 48 Complete exercises #4 – 6.

4) Find a cubic P(x) with zeros 2, 3i, –3i. P(x) = (x – 2)(x2 + 9) P(x) = x3 – 2x2 + 9x – 18

5) Find a quartic P(x) with zeros ±2, ±2i. x = ±2i x2 = (±2)2 x2 = (±2i)2 x2 = 4 x2 = –4 x2 – 4 = 0 x2 + 4 = 0 P(x) = (x2 – 4)(x2 + 4) P(x) = x4 + 4x2 – 4x2 – 16 P(x) = x4 – 16

6) Find a quartic P(x) with zeros ± 2, ±5i. P(x) = (x2 – 2)(x2 + 25) P(x) = x4 + 25x2 – 2x2 – 50 P(x) = x4 + 23x2 – 50

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