Roots of Polynomials
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Roots of Polynomials
a quadratic polynomial: [pic] a cubic polynomial: [pic]
a quintic polynomial: [pic]
Polynomial equation in factored form: y = (x + 3)(x + 1)(x - 2)
Example: Write it in standard form: y = (x + 3)(x + 1)(x - 2)
Factoring Review
Factoring Out a Monomial: Undistributing Special Pattern: [pic]
Ex: [pic] Ex: [pic]
Polynomial equation in standard form: y = 2x3 + 10x2 +12x
Example: Write it in factored form:
Check it by multiplying:
Example: Write it in factored form: y = 3x3 - 3x2 -36x
Example: Write it in factored form: y = 75x3 - 20x
Zeros/Roots/x-intercepts/solutions
y = x3 + 2x2 - 5x - 6 = (x + 3)(x + 1)(x - 2)
• (x + 3), (x + 1), & (x - 2) are factors of x3 + 2x2 - 5x - 6.
• -3, -1, 2 are solutions of x3 + 2x2 - 5x - 6 = 0.
• Plugging each of these numbers in for x makes equation true.
• -3, -1, 2 are x-intercepts of the graph of y = x3 + 2x2 - 5x - 6.
• When x = -3, x = -1, or x = 2, the graph is on the x-axis.
• -3, -1, 2 are zeros of h(x) = x3 + 2x2 - 5x - 6.
• When the graph is on the x-axis, the y-value - or function value - equals zero.
• -3, -1, 2 are roots of h(x) = x3 + 2x2 - 5x - 6.
To find all of these: set y = ______ and ________________.
Examples: Find the zeros of the functions.
f(x)=(x-3)(x+4)(x-1)
g(x) = 2x3 + 10x2 +12x
h(x) = 75x3 - 20x
Intro to Graphing Polynomial Functions
Some graphing calculator instructions:
To enter an equation:
Press the [pic] button to get to the screen to the top right
Use the [pic] button for the variable x.
Use the parenthesis buttons for parenthesis.
Use the ^ button for exponents.
To see the graph:
Press the GRAPH button to get to the third screen shown.
To adjust the window:
If the window on your graph does not count 10 in each direction, you can reset the window by pressing the ZOOM button and choosing option 6:ZStandard.
To see the table:
Press the 2ND button then GRAPH.
(See TABLE written above the graph button.)
You should be able to scroll up and down with the arrow buttons to see different x-values and the y-values that go with them. It should look something like the third screen shot. If it does not, you will have to reset the table.
To reset the table:
Press the 2ND button then WINDOW. (See TBLSET above the window button.)
Change the settings to match the bottom screen shot. To highlight a word, use the arrow and ENTER buttons.
Some ANY calculator instructions:
To evaluate an equation for a given x-value:
You can “plug in” an x-value to the standard OR factored form. Both should give you the same result.
Ex: To plug in x=-6 to the last example using the standard form. To avoid making errors with a negative, use parenthesis.
Ex: To plug in x=-6 to the last example using the factored form.
Classifying Polynomials
[pic]
Degree (biggest exponent on variable):
Leading term (term with highest degree):
Leading coefficient (coefficient of leading term):
More vocab: see chart p301 in text.
Examples:
Write the polynomial in standard form.
Then classify it by degree and by number of terms.
[pic] [pic]
The greatest value (y-value) of the points in a region of a graph is called a __________________.
(Think of the top of a hill.)
The least value (y-value) of the points in a region of a graph is called a __________________.
(Think of the bottom of a valley.)
Multiplicity of a Zero
A repeated zero is called a __________________. A multiple zero has a ____________________ equal to the number of times the zero occurs.
Example: Find the zeros of the function. State the multiplicity of multiple zeros.
[pic] [pic]
Examples:
Write a polynomial function in standard form with zeros at -2 and 3 (multiplicity 2).
Write a polynomial function in standard form with zeros at -4, -2, and 0 (multiplicity 3).
Graphing Polynomials on the TI-84
To find minimum and maximum:
2ND TRACE
3: minimum
4: maximum
[pic] [pic] [pic] [pic]
Example: Graph g(x) = x4 - 7x3 + 12x2 + 4x - 16 = (x + 1)(x - 2)2(x - 4)
Before we graph what x-intercepts and y-intercepts do we expect?
Graph using calculator. Do we need to change window?
ZOOM 6:ZStandard WINDOW
[pic] [pic]
Example: Graph h(x) = -2x4 + 3x3
x-intercepts and y-intercepts: end behavior: Shape we expect:
Change window?
[pic] Relative minimums and maximums:
Example: Graph f(x) = 10x3 + 5x2 - 40x - 20
x-intercepts and y-intercepts: end behavior: Shape we expect:
Change window?
[pic] Relative minimums and maximums:
Example: Graph g(x) = x3 + 10x2 - 25x - 250
x-intercepts and y-intercepts: end behavior: Shape we expect:
Change window?
[pic] Relative minimums and maximums:
Dividing Polynomials
Divide x2 + 3x - 12 by x - 3 Divide x3 + 2x2 - 5x - 6 by x+3
Is x - 3 a factor? Is x+3 a factor?
Divide x3 + 2x2 - 5x - 6 by x2 - x - 2 Divide x3 + 1 by x + 1
Is it a factor? Is it a factor?
Synthetic Division
Review Long Division Synthetic Division
3x3 - 4x2 + 2x - 1 3x3 - 4x2 + 2x - 1
x – 1 x - 1
(5x3 - 6x2 + 4x - 1) ÷ (x - 3) (x4 - 5x2 + 4x + 12) ÷ (x + 2)
Factor (x3 - 13x + 12).
Hint: Here is what the graph of y = x3 - 13x + 12 looks like.
Factor (x3 - 6x2 + 3x +10).
Hint: One factor is (x - 5).
Use Synthetic Division to Factor (x3 - 4x2 - 3x +18).
Use Synthetic Division to Factor (x4 + 6x3 + 8x2).
3x2 + 5x + 2
3x +2
Use Synthetic Division to Factor (x3 + 12x2 + 47x + 60).
Using Synthetic Division to Evaluate a Polynomial Equation
f(x) = x4 + 3x3 – x2 – 3x + 5
x y
-6
-5
-4
-3
-2
-1
-0.5
0
0.5
1
2
3
4
5
6
We can find f (a) using Synthetic Division.
Put ____ in the box.
The ______________ = f (a)
Use Synthetic Division to find the following for the above function.
a = -6
a = -2
a = 0
a = 1
Special Factoring Patterns
Pattern we already know: [pic]
Example: [pic]
By the way: [pic] is NOT factorable.
Example: [pic]
New patterns: [pic]
[pic]
Examples: [pic]
[pic]
More Factoring and Solving by Factoring
Find the zeros...
x4+3x2-10 = y x5+3x3-10x = y
x3+ 27= y 375x5+ x2= y
Solving Polynomials by Graphing
Check out the Resources page of my website: purtle. for links to free graphing software.
[pic]
[pic] [pic]
What if one side has a zero?
Example: Solve [pic]
Sometimes the solution isn't always an integer.
Example: Solve [pic]. Round to the nearest hundredth.
Notes to help me graph:
-----------------------
Hint:
Multiply
things at a time.
[pic]
Hint:
Sometimes you will have to use more than one factoring technique.
[pic]
[pic]
-222.8
-76.8
-17.01
0
1.05
0
1.35
3.84
4.05
0
[pic]
[pic]
IF (x - 1) DOES end up being a factor,
what would the associated zero be?
That is the number that goes in the box.
IF (3x + 2) DOES end up being a factor,
what would the associated zero be?
That is the number that goes in the box.
Graph each of these and find the ______________ where they ________________.
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