Unit 10 Notes - Miss Seitz's Online...
Transcript of Unit 10 Notes - Miss Seitz's Online...
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Unit 10 Guided Notes Polynomial Functions Part 2
Standards: A.APR.2, A.APR.3, A.REI.4b, A.REI.11, A.SSE.3a, F.IF.7c, F.IF.8a
Swartz Creek High School – Algebra 2A
Name: _____________________________________________ Period: ____________________
• Miss Seitz’s tutoring:
By appointment. See Miss Seitz.
• Website with all videos and resources
www.msseitz.weebly.com
Miss Kari Seitz
Text: 810.309.9504
Classroom: 810.591.1829
Email: [email protected]
Concept #
What we will be learning...
Text
Basics for Factoring Polynomials
⃣ Identify and factor out the Greatest Common Factor (GCF) of a polynomial function
⃣ Determine if a binomial is a factor of a polynomial
⃣ Factor by grouping
6.4
Factoring Polynomials
⃣ Factor trinomials where a=1
⃣ Use the difference of squares
⃣ Factor trinomials where a ≠ 1
6.4
Finding Real Roots of Polynomials
⃣ Find real roots by factoring
⃣ Identify multiplicities
6.5
Writing Polynomial Functions
⃣ Write the simplest polynomial given the roots 6.6
Graphs of Polynomial Functions
⃣ Identify key features of a graph of a polynomial function
⃣ Identify the end behavior of a polynomial function
⃣ Draw a rough sketch of a polynomial function
6.7
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Basics for Factoring Polynomials Text: 6.4
⃣ I can identify and factor out the Greatest Common Factor (GCF) of a polynomial function.
⃣ I can determine if a linear binomial is a factor of a polynomial.
⃣ I can factor by grouping.
Vocabulary:
Determine if a Linear Binomial is a Factor
The F____________________ T______________ states: For any polynomial, P(x),
(__________) is a factor of P(x) if and only if P(a) = _____________
Example 1: (x + 1) ; P(x) = x2 – 3x + 1
Identifying and Factoring out the GCF
When factoring, always remember to take out the ___________ (Greatest
Common Factor).
Example 2: Factor out the GCF of 6x4 – 12x3 + 4x2 – 18
Factoring by Grouping
Example 3: Factor 2x3 – 2x2 – 50x + 50
Steps:
1. Factor out the GCF (if
possible)
2. Group the first two terms
and the last two terms
3. Find the GCF of each group
4. Write your final answer
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You Try It!
1. Determine if the linear binomial (x + 2) is a factor of 3x4 + 6x3 – 5x – 10
2. Factor out the GCF of 125n3 + 175n2 + 50n + 70
3. Factor 12n3 – 30n2 + 84n – 210 by grouping
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Factoring Polynomials Text: 6.4
⃣ I can factor trinomials.
⃣ I can factor using the sum and difference of cubes.
Vocabulary: Sum of Cubes, Difference of Cubes
Factoring Trinomials (Use the methods from Unit _____)
Example 1: Factor 24m5 – 20m3 – 16m
Sum and Difference of Cubes
If the sign is P_____________________, the pattern is ______ , ______ , ______
If the sign is N_____________________, the pattern is ______ , ______ , ______
Example 2: Use the sum or difference of cubes to factor 2x5 – 16x2
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You Try It!
1. Factor x5 + 9x3 + 18x
2. Use the sum or difference of cubes to factor
A. 27u3 + 125 B. 8m3 - 1
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Finding Real Roots of Polynomials Text: 6.5
⃣ I can find real roots by factoring polynomial functions.
⃣ I can identify multiplicities.
Vocabulary: Roots of an equation, Multiplicity
Definitions
R_____________ of an Equation: Any value of the V_________________ that
makes the equation true.
To identify how many roots a polynomial has, look at its D_____________.
If a polynomial has degree ______ , it has _______ roots
M____________________: The number of times a root of a polynomial
appears as a F__________________ of the polynomial
When a real root has E___________ multiplicity, the graph touches the
x-axis, but does not cross it.
When a real root has O_______ multiplicity greater than _____ , the graph
B____________ as it crosses the x-axis.
Identifying Roots and Factors
FACTORS ROOTS
3
(x – 4)
(x + 5)
–2
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Finding Roots by Factoring
Example 1: Find the roots of 4x6 + 4x5 = 24x4
Steps:
1. Set the polynomial equal to zero
(get all terms on the same side)
2. Factor out the ________
3. Determine the factoring
technique based on the number
of terms and degree
4. Factor
5. Set each factor = to _______
Roots: _______________________________
Multiplicities: _______________________
REMINDERS
When factoring a polynomial equation, always remember to take out the
_________ (greatest common factor).
Look at the number and degree to decide which factoring method to use.
Refer to previous sections of notes for factoring techniques.
When solving a polynomial equation, set each F_________________ equal to ____
When identifying multiplicities, determine how many times the identical
root appears as a S____________________________
You Try It!
Solve each polynomial by factoring.
1. 2x6 – 10x5 – 12x4 = 0
Roots: _________________________________
Multiplicities: _________________________
2. x3 – 2x2 – 25x = –50
Roots: _________________________________
Multiplicities: _________________________
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Writing Polynomial Functions Text: 6.6
⃣ I can write the simplest polynomial given the roots.
Vocabulary:
Writing the Simplest Polynomial
In this section, you will write polynomial functions given two or more R_______.
Roots may be R____________________, I________________________, or C______________
REMEMBER: Irrational and complex zeros come in P__________. This means if
you see 3 + 4i as a zero, then _________________ is also a zero, even if it is not
listed. If you see √3 as a zero, then _______________ is also a zero.
THIS IS NOT THE CASE FOR REAL NUMBERS
Example 1: Write the simplest polynomial given the roots for each below.
A. –2, 3, 4
B. 2i, 1, –2
C. 5, √7
You Try It!
Write the simplest polynomial given the roots below.
1. 3, 5, -1 2. 3i, -4
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Graphs of Polynomial Functions Text: 6.7
⃣ I can identify key features of a graph of a polynomial function.
⃣ I can identify the end behavior of a polynomial function.
⃣ I can draw a rough sketch of a polynomial function.
Vocabulary: End Behavior
End Behavior
End Behavior: The description of the values of the function as x approaches
positive infinity (_________) or negative infinity (________)
The D_____________ and L_______________ Coefficient of a polynomial function
determine its E_________ B_________________________.
Each polynomial graph has a distinctive shape based on its D__________.
When determining End Behavior, only look at the term with the
H__________________ D_____________.
If the Leading Coefficient is:
• P________________: The arrows on the right side of the graph will point ____
• N________________: The arrows on the right side of the graph will point ____
If the Degree is:
• E__________: The arrows will point in the S__________ direction
• O_________: The arrows will point in O__________________ directions
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Determining End Behavior
Example1: Find the end behavior of the following:
A. 2x5 + 3x2 – 4x - 1
Lead Coefficient: _______
Degree: _______
End Behavior:
As x _________ P(x)_________
As x _________ P(x)_________
B. Lead Coefficient (+ or – ) : _______
Degree (even or odd) : _______
End Behavior:
As x _________ P(x)_________
As x _________ P(x)_________
C. Lead Coefficient (+ or – ) : _______
Degree (even or odd) : _______
End Behavior:
As x _________ P(x)_________
As x _________ P(x)_________
Sketching a Rough Graph
Example 2: Graph f(x) = x3 – 7x2 – 6x + 72 which is factored as (x–6)(x+3)(x–4)
Lead Coefficient (+ or – ) : _______
Degree (even or odd) : _______
End Behavior:
As x _________ P(x)_________
As x _________ P(x)_________
x-Intercepts: ________________________
y-Intercept: _________________________
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You Try It!
1. Determine the end behavior of the function f(x) = 3x12 + 2x8 – 9x3 + 4
Lead Coefficient (+ or – ) : _______
Degree (even or odd) : _______
End Behavior:
As x _________ P(x)_________
As x _________ P(x)_________
2. Graph the function f(x) =x3– 4x2–7x+10 which is factored as (x+2)(x–5)(x–1)
Lead Coefficient (+ or – ) : _______
Degree (even or odd) : _______
End Behavior:
As x _________ P(x)_________
As x _________ P(x)_________
x-Intercepts: ________________________
y-Intercept: _________________________