Name:
Date:
Period:
Polynomials
Lessons: 6-1, 6-4, 6-5
Packet 7
Tennessee State Standards
3103.3.2 Determine roots of a higher order
polynomial.
Common Core State Standards
A-SSE- 2 Use the structure of an expression to
identify ways to rewrite it. For example, a
difference of two squares
A-APR-2 Know and apply the Remainder Theorem:
For a Polynomial p(x) and a number a, the
remainder on division by x – a is p(a), so p(a) = 0 if
and only if (x – a) is a factor of p(x).
A-APR-3 Identify zeros of polynomials when
suitable factorizations are available, and use the
zeros to construct a rough graph of the function
defined by the polynomial.
Name:
Lesson 6-1
Date:
Period:
Polynomials
p.1
Degree
Leading Coefficient
Polynomial Terms
Degree
Leading Coefficient
Binomial
Trinomial
Polynomial
Classifying Polynomials by Degree
Name:
Date:
Lesson 6-1
Polynomials
Period:
p.2
Classifying Polynomials
2x + 4x3 – 1
1. Write terms in descending order (based on
degree).
2. Identify the Leading Coefficient
3. Identify the degree
4. How many terms?
5. Classify/give a name
Part A: Guided Practice
1. Write terms in
descending
order (based on
degree).
2.
Identify the
Leading
Coefficient
3. Identify the
degree
4. How many
terms?
5. Classify/give a
name
4x + 2x2 – 7 + x3
5x2 – 4x3
4x4 + 8x2 + 1 – 3x
Name:
Date:
Lesson 6-4
Period:
Factoring Polynomials
p.3
Determining Whether a Binomial is a Factor of a Polynomial
1. To determine if (x – 3 ) is a factor,
use substitution. If f(3) = 0, (x – 3)
IS a factor. If f(3) = anything else,
it is not.
Is (x – 3) a factor of f(x)= x2 + 2x + 3?
2. Use a calculator to substitute 3
for x in f(x).
3. What is the result? Is (x – 3) a
factor?
1. If f(-4)= 0, (x + 4) is a factor. If f(4) = anything else, it is not.
2. Substitute -4 for x in f(x). **If using
a calculator, be sure to put -4 in
parentheses (-4) each time! Why is this
important?
3. What is the result? Is (x + 4) a
factor?
Is (x+4) a factor of f(x) = 2x4 + 8x3 + 2x + 8?
Name:
Date:
Lesson 6-4
Factoring Polynomials
Period:
p.4
Review: Factoring a Difference of Two Squares
1. Recognize that each of the two
terms is a perfect square.
Coefficients are perfect squares
and exponents are all even.
There is a minus sign (-) in
between, making it a difference.
25x4 - 81
2. Take the square root of the first
term.
3. Take the square root of the
second term.
4. One factor is (a + b), the other is
(a – b).**These are called conjugates.
Other Popular Factoring Methods
Name
What it looks like
Method
Name:
Date:
Lesson 6-4, 6-5
Factoring Polynomials
Period:
p.5
Part A: Guided Practice
Factor.
1. 16- 100x6
2. 5x4 + 40x
3. 8x3 – 27
Finding the Roots of a Polynomial Equation
*A root is another name for a ________________________.
1. Factor out a GCF, if possible.
2. Factor the trinomial or binomial.
3. Set each factor = 0.
4. Solve for x.
5. These are the roots.
6. Check using the calculator.
- Graph the function
- Calc (2nd Trace)
- #2 Zero
- Left Bound, Right Bound, Guess
2x3 + 4x2 – 30x
Name:
Lesson 6-5
Date:
Roots of Polynomials
Period:
p.6
Part A: Guided Practice
Find the roots (solve).
Find the roots (solve).
3x4 – 6x2 – 24
X4 – 13x2 = -36
Name:
Date:
Lesson 6-5
Roots of Polynomials
Period:
p. 7
Multiplicity
 Multiplicity is the _________ of times a _______ occurs as a factor.
 Even multiplicity means that the graph ____________ the x-axis.
 Odd multiplicity means that the graph ____________ the x-axis.
1. Look at the graph of x3 – 9x2 + 27x – 27. The root is 3. What does the graph
do at x = 3? _____________
Draw a rough sketch of the graph.
2. Factor to find the roots of this polynomial. State what the graph does at
each of those roots based on the multiplicity.
f(x) = 4x6 + 4x5 – 24x4
Name:
Date:
Lesson 6-5
Roots of Polynomials
Period:
p. 8
The Rational Root Theorem
This theorem helps you list all the
4x4 – 21x3 + 18x2 + 19x – 6 = 0
POSSIBLE factors of a polynomial.
1. Write all the factors of the
constant term. (positive and
negative)
2. Write all the factors of the
leading coefficient. (positive
and negative)
3. Divide every number on the
first list by every number on
the second list.
4. Simplify each number.
5. Cross out duplicates.
6. These are the possible roots!
Name:
Lesson 6-5
Date:
Roots of Polynomials
Period:
p. 9
*Now we will use our list and find out which possible roots really are roots.
1. Copy down the list of roots from
the previous page.
2. Write down the polynomial.
3. Begin using substitution to test
each of the roots. When one
results in 0, it is a root!
4. To check, graph the polynomial!
Name:
Lesson 6-5
Date:
Roots of Polynomials
Period:
p.10
Part B: Homework
1. Identify the leading
coefficient, degree, and
number of terms. Name
the polynomial.
5x2 – 4x3
2. Verify whether or not
(x – 2) is a factor of
x2 + 3x – 7. Show your
work.
3. Verify whether or not
(x – 8) is a factor of
x5 – 8x4 + 8x – 64. Show
your work.
4. Factor x6 – 1
5. Factor 40 – 5x3
6. List all the possible roots
x4 + 3x3 – 3x2 – 12x – 4
7. List all the possible roots
x3 + x2 – 2x – 8 = 0
8. Find all the real roots by
factoring. State the
multiplicity of each.
x5 – 2x4 – 24x3 = 0
9. Find all the real roots by
factoring. State the
multiplicity of each.
3x5 + 18x4 – 21x3 = 0
10. List all the possible roots
AND test each one using
substitution. Show your
work.
x4 – 3x2 – 4 = 0
Show your work for #10 here.