1

1

1

1

1

1

2

3

3

1

1

1

4

4

6

Final Equation

Multiply

y=(x-(2+4i))(x-(2-4i))(x-6)

y=((x-2)-4i)((x-2)+4i)(x-6)

y=(x-2)(x-2)(x-6)+4

y=(x -4x+4)(x-6)+4

y=x -6x -4x -16x+4x-24+4

Simplify

y=x -6x -4x -16x+4x-24+4

y=x -10x -12x-20

Writing an Equation with the Roots

The Given Roots

are 6, 2+4i.

3

y=(x-(2+4i))(x-(2-4i))(x-6)

Write an equation using the factors found using the roots.

If one of the roots is (2+4i), then another root has to be (2-4i), which is its complex conjugate. There cannot be an odd number of imaginary roots.

2

2

2

3

2

2

2

3

3

y=x -10x -12x-20

3 2

Conjugate: Number pairs in the form of a+ b and a- b are conjugates. Number pairs in the form a+bi and a-bi are complex conjugates. 8+ 5 and 8- 5 are conjugates. 7+3i and 7-3i are complex conjugates. The products of conjugate pairs are rational numbers.

Degree: The degree of a polynomial is the largest degree of any term in the polynomial. In the equation y=17x +9x +2,

4

is the degree.

Multiplicity: The multiplicity is the number of times the same root occurs.

4

2

Root

-5 has a

multiplicity

of 2

Finding the Roots of a

Polynomial Equation Using ...

... a Calculator

Step 1:

Type your equation into the calculator.

Put your equation into Y1 and 0 into Y2.

Step 2:

Graph your equation.

Step 3:

Press 2nd Trace and select

5: intersect.

Step 4:

Press enter.

Step 5:

Press enter again.

Step 6:

Press enter one more time.

Step 7:

The x value of the intersection

with the x-axis (y=0) is one root.

Repeat two more times to find the rest

of them.

Answer:

The roots of y=x -39x-70 are -5, -2, and 7.

... Synthetic

Division

Step 1:

Using your calculator, find one root

of your equation.

Equation: y=x -39x-70.

Root: -2

Step 2:

Divide the coefficients of the equation by the root you already found. Put zeros as place holders for terms that have coefficients of zero.

Equation: y=x -39x-70 or y=x -0x -39x-70

Root: -2

Step 3:

Write a new equation using the

new coefficients. It should be one degree smaller than the original equation.

Then factor.

y=x -2x-35.

y=x -7x+5x-35

y=(x-7)(x+5)

Step 4:

The factors of an equation

are (x-root). Find the remaining roots

using the factors.

y=(x-7)(x+5)

Answer:

The remaining roots are 7 and -5.

3

3

3 3 2

-2 1 0 -39 -70

-2 4 70

1 -2 -35 0

-2 1 0 -39 -70

-2 4 70

1 -2 -35 0

2

2

... the Quadratic

Formula

The quadratic formula is -b+ b -4ac

The quadratic equation 2a

is y=x -2x-35, so a=1,b=-2, and c=-35.

-(-2)+ (-2) -4(1)(-35) =7

2(1)

-(-2)- (-2) -4(1)(-35) =-5

2(1)

2

2

2

(x+y)

(x+y)

(x+y)

(x+y)

(x+y)

0

1

2

3

4

Expand (4x+3) to degree 4:

The coefficients would be 1,4,6,4,1.

1(4x) (3)+4(4x) (3)+6(4x) (3)+4(4x) (3)+1(4x) (3)

1(256x )+4(192x )+6(144x )+4(108x)+1(81)

256x +768x +864x +432x+81

4

4 0 3 1 2 2 1 3 0 4

4 3 2

4 3 2

A square pool that is as deep as its length

is surrounded by deck that is x feet wide and

20 feet long. How much water can the pool hold?

(-2x+20) Coefficients according to Pascal's Triangle: 13 31

1(-2x) (20) +3(-2x) (20) +3(-2x) (20) +1(-2x) (20)

1(-8x )+3(80x )+3(-800x)+1(8000)

Answer:

-8x +240x -2400x+8000 ft

20ft

20ft

x

x

3

3 0 2 1 1 2 0 3

3 2

3 2 3

http://www.presentationhelpdesk.com/your-millionaire-game-1236.html

**Practical Application Problem**

Cube A has a lenth of x. Cube B's volume is 900ft, which is the sum of three plus cube A's volume squared. What is the value of x?

(x +3) =900

x +6x +9=900

x +6x -891=0

(x -27)(x +33)

x = 27 3 = -33

x=3 ft

Plant Growth

Amount of Sunlight

(hrs perday)

Daily Growth (cm)

0

2

4

6

8

10

12

14

16

0

1.3

11.9

2.7

4.0

7.4

15.5

13.2

9.6

Quadratic Model: -.0524x +1.7348x-1.84

r =.8264

Cubic Model: -.0196x +.4183x -1.1052x+.796

r =.9669

Quartic Model: -.0017x +.0352x -.1265x +.6008x+.1386

r =.9846

Quartic is the best fit model.

How tall will the plant grow with 17 hours of sunlight?

Answer: 3.6cm

3 2

6 3

6 3

3 3

3

3

3

3

2

3 2

2

2

2

4 3 2

Finding the

Relative Maximum

Equation: y=-x +5x -x-5

3 2

2. Go to 2nd trace and

select 4: maximum.

3. Press enter three times.

1. Plug the equation in

the calculator next to Y1

and 0 next to Y2.

2. Graph it, then select

5: intercect in 2nd trace.

3. Press enter three times. Repeat two

more times to find all of the zeros.

**Polynomials**

By Nadia Johnson

**Words to Know**

2

Pascal's Triangle

Application Problem

**Lesson on Pascal's Triangle**

Best Fit Model

1. Plug the equation into a

calculator and graph it.

The relative maximum is

(-3.43, 16.9)

Finding the

Relative Minimum

Equation: y=-x +5x -x-5

3 2

2. Go to 2nd trace and

select 3: minimum.

3. Press enter three times.

1. Plug the equation into a

calculator and graph it.

The relative minimum is

(.10, -5.05)

3

Finding the

Zeros

Equation: y=-x +5x -x-5

3 2

The zeros are -5,-1, and 1.

**Game for Practice on**

Polynomial Functions:

Polynomial Functions: