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: kseitz@swcrk.org

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: _________________________