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Polynomials

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by

Nadia J.

on 25 December 2011

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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:
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:
Full transcript