6-3
Polynomials
Going Deeper
Essential question: What parts of a polynomial represent terms, factors,
and coefficients?
TE ACH
Standards for
Mathematical Content
1
A-SSE.1.1 Interpret expressions that represent a
quantity in terms of its context.*
A-SSE.1.1a Interpret parts of an expression, such
as terms, factors, and coefficients.*
A-SSE.1.1b Interpret complicated expressions
by viewing one or more of their parts as a single
entity.*
ENGAGE
Questioning Strategies
• In the polynomial 4x 3 + 2x - 9, is the constant
term 9 or -9? Explain. -9; To determine what
the terms are you must rewrite the polynomial
as a sum, 4x 3 + 2x + (-9), which shows that the
constant term is -9.
• A student claims that the degree of the
polynomial 9q + 6 - 3q2 is 1 because the first
term is 9q and the exponent of q is 1. Why is this
not correct? What is the correct degree?
Vocabulary
constant term
degree
leading coefficient
parameter
standard form
variable term
The degree of a polynomial in one variable is the
greatest power of the variable, even if it does
not appear in the first term of the polynomial.
The correct degree is 2.
Variables and Expressions
Math Background
Students have learned the difference between a
numerical expression and an algebraic expression.
The terms of a numerical expression are numbers,
while the terms of an algebraic expression can be
numbers, variables, or a product of numbers and
variables. Like terms, which have identical variable
parts, can be combined to simplify an algebraic
expression.
2
EXAMPLE
Questioning Strategies
• What is the degree of any constant term?
The degree of a constant is 0.
IN T RO DUC E
• If the polynomial 16y + 2y4 + 11 - 3y2 contained
a y3 term, where would it appear when you write
the polynomial in standard form?
It would be between the 2y4 and -3y2 terms.
Give students an algebraic expression such as:
13x + 7 - 12x + 4.
Ask students a series of questions about the parts of
the expression such as the following:
EXTRA EXAMPLE
Write the polynomial 3z - 9z3 - 8z5 + 2z2 in
standard form.
-8z5 - 9z3 + 2z2 + 3z
• What are the terms of the expression? Do any of
the terms have coefficients? If so, which ones?
• Are there any like terms in the expression? If so,
which ones?
• How do you simplify an algebraic expression?
Can this expression be simplified? Explain how.
Chapter 6
337
Lesson 3
© Houghton Mifflin Harcourt Publishing Company
Differentiated Instruction
The displayed expressions in Engage 1 use callouts
to point out the different parts of the expression.
Encourage students, especially visual learners, to
identify the parts of the polynomials in the Reflect
exercises in a similar way. For example, students
can circle the terms and draw rectangles around the
coefficients in the variable terms.
Prerequisites
Name
Class
Notes
6-3
Date
Polynomials
Going Deeper
Essential question: What parts of a polynomial represent terms, factors, and
coefficients?
A-SSE.1.1a
1
ENGAGE
Investigating Parts of a Polynomial
The parts of a polynomial that are added are called the terms of the polynomial. Each
term is either a constant term (a number) or a variable term (a variable, or a product of
one or more variables with whole number exponents). Terms may have factors that are
numbers, variables, or combinations of both. To identify the terms of the polynomial
4x3 + 2x - 9, first rewrite the subtraction as addition.
variable terms
constant term
factors of 4x3: 1, 2, 4, x, x2, x3, 2x, 2x2, 2x3, 4x, 4x2, 4x3
factors of 2x: 1, 2, x, 2x
4x3 + 2x + (-9)
coefficient
variable part
Variable terms in a polynomial can be broken down into a coefficient and a variable part,
as shown above for the first term, 4x3. For the second term, 2x, 2 is the coefficient and x is
the variable part. The degree of a polynomial in one variable is the greatest power of the
variable. For the polynomial above, the degree is 3. Notice that the variable terms have the
common factors 2, x, and 2x, but there are no common factors greater than 1 of all three terms.
You can also have a polynomial in a variable other than x. Below is a polynomial in the
variable q.
© Houghton Mifflin Harcourt Publishing Company
9q + 6 - 3q2
Variable terms: 9q, -3q2
Constant term: 6
Degree: 2
REFLECT
1a. What is the degree of the polynomial 8y3 + 6y - 4y5 + 2y2? Do the coefficients of
the terms have any common factors? Do the variable parts of the terms have any
common factors? What is the greatest common factor of all four of the terms of the
polynomial?
The degree is 5. Yes, 2 is a common factor of the coefficients. Yes, y is a common
factor of the variable parts. The greatest common factor of all the terms is 2y.
1b. The polynomial x4 + x3y4 + xy5 has two variables. The degree of any term is the sum
of the exponents of its variables. The degree of the polynomial is the degree of the
term with the greatest degree. Find the degree of each term of this polynomial and
the degree of the polynomial.
degree of x4: 4; degree of x3y4: 7; degree of xy5: 6; degree of polynomial: 7
337
Chapter 6
Lesson 3
Writing polynomials in standard form lets you easily identify the characteristics of and
compare different polynomials. Polynomials in one variable are written in standard form
when the terms are in order from greatest degree to least degree. Standard form also
makes it easy to compare the leading coefficients of two polynomials, or the coefficients of
their first terms.
A-SSE.1.1b
EXAMPLE
Writing Polynomials in Standard Form
Consider the polynomial 16y + 2y4 + 11 - 3y2.
A
4
The polynomial has
1
16y (degree
(degree
2
),
11
(degree 4),
(degree 0), and -3y2
). The term with the highest degree is
polynomial has degree
B
terms. The terms and their degrees are
2y4
4
2y4
, so the
.
4
2
The polynomial written in standard form is
2y - 3y + 16y + 11
The leading coefficient of the polynomial is
2
.
REFLECT
2a. Notice that the number of terms is the same as the degree of the polynomial in the
Example. Is this true for any polynomial? Explain.
No; the number of terms and the degree of a polynomial are not
always the same. Consider 5x2 + 7x + 4, which has 3 terms and a
degree of 2.
2b. A trinomial is a polynomial that has three terms. What possible degrees can a
trinomial have (consider that each term has a different exponent and that the
exponents are whole numbers)? Explain.
The least degree is 2, because to have three terms, the lowest exponents must be
0, 1, and 2. Any degree above 2 is also possible, since you do not have to have
terms of every degree. For example, x4 - x2 + 1 and x100 + x10 + x5 are trinomials.
Polynomials often contain letters other than the variable(s) that take specific values
depending on the particular situation. These values are often referred to as parameters.
For ax + b, a general linear polynomial in the variable x, the letters a and b are parameters.
The parameters a = 50 and b = 100 give 50x + 100. This might represent, for example, the
total cost of a gym that has a monthly cost of $50 and a membership fee of $100.
© Houghton Mifflin Harcourt Publishing Company
© Houghton Mifflin Harcourt Publishing Company
2
The quadratic polynomial -__21at2 + v0t + h0 is a polynomial in the variable t, for time.
It represents the height of a projectile launched upward in the presence of gravity. The
parameters a, v0, and h0 represent the acceleration caused by gravity in a location, the
initial upward velocity, and the initial height. Once you know the parameters for a given
situation, you can find the heights for different values of the variable t.
Chapter 6
Chapter 6
338
Lesson 3
338
Lesson 3
3
CLOS E
EXAMPLE
Questioning Strategies
• What coefficient, if any, would not change if a
different model rocket were launched from a
different height with a different launch speed?
Explain. -16; the coefficient -16 depends on the
Essential Question
What parts of a polynomial represent terms, factors,
and coefficients?
The terms of a polynomial are the parts that are
added together. A polynomial may contain constant
terms and variable terms. For variable terms,
the coefficient is the number multiplied by the
variable part of the term. Each number or variable
multiplied in a variable term is a factor of the term.
acceleration of gravity which is a constant. The
other coefficients depend on the rocket.
• How would you write a correct polynomial for
the model rocket in Reflect question 3b? Give the
correct polynomial. Change 9 inches to 0.75 feet;
-16t 2 + 350t + 0.75
Summarize
Have students make a graphic organizer to show
how they can determine the parts of a polynomial.
A sample is shown below.
EXTRA EXAMPLE
A ball is dropped from a height of 37 feet above
the ground. The height of the ball t seconds after
it is dropped can be modeled by -16t 2 + 37.
Interpret each term in the polynomial. The term
-16t 2 depends on the acceleration of gravity times
Highlighting
the Standards
the square of the time in seconds after the ball is
dropped. The term 37 is the height (in feet) from
which the ball was dropped.
Teaching Strategy
Before beginning 3 EXAMPLE , it may be helpful
to discuss projectile motion. Tell students that a
projectile is an object that is propelled into the air
by a force, but the force is no longer applied while
the object is in the air. This causes the object to fall
to the ground. The height (in feet) of a projectile t
seconds after the object is projected into the air can
be modeled by -16t2 + v0t + h0 , where v0 is the
initial vertical velocity (in feet per second) of the
object and h0 is the initial height (in feet) of the
object.
PR ACTICE
Where skills are
taught
What are the
terms of the
polynomial?
Chapter 6
Which terms
are variable
terms?
2 EXAMPLE
EXS. 1–6
3 EXAMPLE
EX. 7
What are the
factors of each
variable term?
Which terms
are constant
terms?
339
Where skills are
practiced
What is the variable
part of each term?
What is the coefficient
of each term?
Lesson 3
© Houghton Mifflin Harcourt Publishing Company
This lesson provides opportunities to address
Mathematical Practices Standard 7 (Look
for and make use of structure). By analyzing
polynomials, students learn how to identify
the terms and what they represent. The factors
of each variable term are used to identify the
coefficient and the variable part of the term.
Understanding this structure is essential to
interpreting the terms of a polynomial used
to model a real world situation. For example,
students learn that the coefficient of the term
-16t 2 in the polynomial in 3 EXAMPLE
depends on the acceleration of gravity.
Students can use this knowledge to interpret
the term -4.9t 2 in the polynomial in Exercise 7.
3
Notes
A-SSE.1.1
EXAMPLE
Interpreting Polynomials
A compressed air model rocket is launched straight into the air from a platform
3 feet above the ground at an initial upward velocity of 350 feet per second.
The height of the rocket in feet t seconds after being launched is modeled by
-16t2 + 350t + 3. Interpret each term in the polynomial. Then use unit analysis
to show how the value of the polynomial is a distance in feet.
A
The terms of the polynomial are
-16t2, 350t, and 3
.
2
The term -16t depends on the acceleration of gravity times the
square
of
the time in seconds after the rocket is launched.
is the initial upward velocity
The term 350t
times
the time in seconds after the rocket is launched.
The term
3
.
is the
height in feet from which the rocket was launched.
B
.
To use unit analysis, write the polynomial with its units. Then divide out any
common units.
2
(
ft · t s
-16 __
2
s
2
) + 350 __fts · t
s + 3 ft = -16t2
(
) (
)
ft · s
ft · s
__________ + 350t __________ + 3
s2
s
ft
= -16t2 ft + 350t ft + 3 ft
The simplified expression shows that the polynomial gives a distance in feet.
© Houghton Mifflin Harcourt Publishing Company
REFLECT
3a. Suppose the model rocket in the Example is launched from a platform 10 feet
above the ground at the same initial upward velocity. Explain how the polynomial
that models the height in feet of this rocket t seconds after the rocket is launched
differs from the polynomial in the Example.
The constant term in the polynomial will change from 3 to 10.
3b. Suppose the model rocket in the Example is launched from a platform 9 inches
above the ground. Does the polynomial -16t2 + 350t + 9 correctly model the
height of the rocket in feet t seconds after it is launched? Explain.
No; the height of the rocket is in feet, so the height of the platform in the
polynomial must also be given in feet, not inches.
339
Chapter 6
Lesson 3
PRACTICE
For each polynomial, find the variable terms and their coefficients, any constant
terms, and the degree of the polynomial. If all the terms of a polynomial have any
common factors greater than 1, find the greatest common factor of the terms.
variable terms: 6x(coefficient 6), -4x3(coefficient -4); constant term: 14;
degree: 3; GCF of terms: 2
2. -12t - 24t3 + 18t2 - 30t4
variable terms: -12t(coefficient -12), -24t3(coefficient -24); 18t2(coefficient 18);
-30t4(coefficient -30); constant term: none; degree: 4; GCF of terms: 6t
3. x3 + 3x2y + 3xy2 + y3
variable terms: x3(coefficient 1), 3x2y(coefficient 3); 3xy2(coefficient 3);
y3(coefficient 1); constant term: none; degree: 3; GCF of terms: 1
Write each polynomial in standard form. Identify the degree and leading
coefficient of the polynomial.
4. 16z + 30z3 - 1 + 2z6
2z6 + 30z3 + 16z - 1; degree: 6; leading coefficient: 2
5. x - x4 + x8 - x6 + x2
x8 - x6 - x4 + x2 + x; degree: 8; leading coefficient: 1
6. 2.2y3 - 1.6 - y5 + 3.4y4
-y5 + 3.4y4 + 2.2y3 - 1.6; degree: 5; leading coefficient: -1
7. A compressed air model rocket is launched straight up into the air from a
platform 1.4 meters above the ground with an initial upward velocity of 107
meters per second. The height of the rocket in meters t seconds after the rocket is
launched is modeled by -4.9t2 + 107t + 1.4.
a. Interpret each term in the polynomial.
-4.9t2 depends on the acceleration of gravity times the square of the time in
seconds after the rocket is launched. 107t is the initial upward velocity times the
© Houghton Mifflin Harcourt Publishing Company
© Houghton Mifflin Harcourt Publishing Company
1. 6x - 4x3 + 14
time in seconds after the rocket is launched. 1.4 is the height in meters from
which the rocket was launched.
b. Use unit analysis to show how the value of the polynomial is a distance in
meters.
m
m
-4.9 __
· (t s)2 + 107 __
s · t s + 1.4 m = -4.9t m + 107t m + 1.4 m
s2
Chapter 6
Chapter 6
340
Lesson 3
340
Lesson 3
ADD I T I O N A L P R AC T I C E
AND PRO BL E M S O LV I N G
Assign these pages to help your students practice
and apply important lesson concepts. For
additional exercises, see the Student Edition.
Answers
Additional Practice
1. 3; 3
2. 2; 2
3. 4; 4
4. 4x 8 + 3x 2 - x - 2; 4
5. 3j 3 - 4j 2 - 50j + 7; 3
6. 5k 4 - 4k 3 + 3k 2 + 6k; 5
7. quadratic binomial
9. quartic polynomial
11. 9
8. quartic trinomial
10. 7
12. 10
13. a. 187.5 m
b. 135.6 m
Problem Solving
© Houghton Mifflin Harcourt Publishing Company
1. 87.92 square centimeters
2. 2 s: 296 feet; 5 s: 500 feet
3. UK: 146.25 feet; US: 194.4 feet
4. h = 0.25: 0.9375 cubic feet;
h = 0.5: 1 cubic foot
5. A
6. G
7. C
Chapter 6
341
Lesson 3
Name
Class
Date
Notes
6-3
© Houghton Mifflin Harcourt Publishing Company
Additional Practice
341
Chapter 6
Lesson 3
© Houghton Mifflin Harcourt Publishing Company
© Houghton Mifflin Harcourt Publishing Company
Problem Solving
Chapter 6
Chapter 6
342
Lesson 3
342
Lesson 3