Unit 10 Notes - Miss Seitz's Online...
Transcript of Unit 10 Notes - Miss Seitz's Online...
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
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
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
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
You Try It!
1. Factor x5 + 9x3 + 18x
2. Use the sum or difference of cubes to factor
A. 27u3 + 125 B. 8m3 - 1
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
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: _________________________
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
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
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: _________________________
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: _________________________