M3D13 Have out: Bellwork: pencil, red pen, highlighter, notebook Complete #1a on the handout. 1) Determine all the zeros and sketch the graph of y = P(x). Label all intercepts. y a) Degree: ___ Zeros: ____________ x Remember: Only graph the real zeros, and make sure the graph fits the criteria that is known for sure.

Any other possibilities out there? 4 Degree: ___ Zeros: ____________ y x (–2,0) (2,0) What are the criteria we know? (1) x–intercepts (0, –100) (2) y–intercept (3) end behavior Any other possibilities out there? Draw a 4th degree polynomial that fits these criteria.

3 b) Degree: ___ Zeros: ____________ End behavior: y (0,0) x (7,0) (–7,0) (7,0) End behavior:

Any other possibilities out there? c) 4 Degree: ___ Zeros: ____________ y x (–2,0) (2,0) What are the criteria we know? (1) x–intercepts (2) y–intercept (0, –36) (3) end behavior Any other possibilities out there? Draw a 4th degree polynomial that fits these criteria.

d) 3 Degree: ___ Zeros: ____________ End behavior: y (0, 8) x (–4,0) End behavior: Be sure to put an inflection point somewhere.

e) 4 Degree: ___ Zeros: ____________ End behavior: y (0, 25) (–1,0) (1,0) x (–5,0) (5,0) End behavior:

f) 5 Degree: ___ Zeros: ____________ End behavior: y (0,0) x (4,0) (–3,0) (4,0) End behavior:

Remember: When there is a complex zero there is always a _____________. conjugate pair For instance: –5i is paired up with +5i. (9 + 4i) is paired up with (9 – 4i).

Part 2: Find a standard (expanded) form equation of a polynomial with the given zeros. a) cubic polynomial with zeros x = 2 and i. There must be a conjugate pair to i : –i So there are 3 zeros: x = 2, ±i x = 2 Any time you see a “±” you have to square both sides. x – 2 = 0 Expand using F.O.I.L. or a generic rectangle. Note: We are finding “a polynomial”, so you do not need to find a specific leading coefficient, “a”.

b) quartic polynomial with zeros x = 3 and degree: 4 All the zeros are: Any time you see a “±” you have to square both sides. Expand using F.O.I.L. or a generic rectangle. Check: This is degree is 4!

c) quartic polynomial with zeros x = 4i and degree: 4 2 zeros here. There has to be two zeros here. Expand using F.O.I.L. or a generic rectangle. Check: This is degree is 4.

d) 5th degree polynomial with zeros x = 1, i, 2i. one real zero 2 zeros 2 zeros Expand using generic rectangles Yep, it’s degree 5! Here is a link to check something like this: poly expand link

Complete the worksheet.

Determine the polynomial function P(x) Determine the polynomial function P(x). Write your answers in factored form. –1, –1, 3 a) zeros: __________ y P(x) = a (x + 1)2 (x – 3) 4 = a(–2 + 1)2(–2 – 3) 4 = a(–1)2(–5) (–2, 4) 4 = a(1)(–5) 4 = –5a x

Determine the polynomial function P(x) Determine the polynomial function P(x). Write your answers in factored form. –4, –1, 3, 3 b) y zeros: _____________ (1, 5) P(x) = a (x +4) (x +1) (x – 3)2 5 = a(1 + 4)(1 + 1)(1 – 3)2 x 5 = a(5)(2)(–2)2 5 = a(5)(2)(4) 5 = 40a