1. U7D7 Have out: Bellwork: x – 4 x2 + 4x + 42 x + 2y x2 – 2xy + (2y)2
Assignment, red pen, highlighter, notebook
U7D7
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

2. 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

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
x4 = 1
What is the degree? 4
How many zeros should we have? 4

4. So what happened to the other 2 solutions? 1) P(x) = x4 – 1
Let’s solve it another way:
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
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.

5. 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
(x + 1)2 = –4
x2 = –4
x2 = 5
–1 –1

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

7. 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
x2 = –1
x = 4

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

9. 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

10. 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”.

11. 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.
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

12. 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.

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

14. 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

15. 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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