Present Remotely

Send the link below via email or IM

• Invited audience members will follow you as you navigate and present
• People invited to a presentation do not need a Prezi account
• This link expires 10 minutes after you close the presentation

Do you really want to delete this prezi?

Neither you, nor the coeditors you shared it with will be able to recover it again.

Polynomials

No description
by

on 25 December 2011

Report abuse

Transcript of Polynomials

Roots: The root of a function is any value of the variable for which the function is zero. When graphed, the x-intercepts (where y=0) represents the roots.
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:
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.
The roots of y=x -39x-70 are -5, -2, and 7.
... Synthetic
Division
Step 1:
Using your calculator, find one root
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)
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
Formula

The quadratic formula is -b+ b -4ac
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)
-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
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?
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
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:
Full transcript