5-1 Practice Operations With Polynomials

Learning Objectives

In this section, yous will:

• Identify the caste and leading coefficient of polynomials.
• Add and subtract polynomials.
• Multiply polynomials.
• Use FOIL to multiply binomials.
• Perform operations with polynomials of several variables.

Maahi is building a little free library (a pocket-size firm-shaped book repository), whose front is in the shape of a foursquare topped with a triangle. At that place will be a rectangular door through which people can have and donate books. Maahi wants to find the area of the forepart of the library so that they can buy the correct amount of paint. Using the measurements of the front of the house, shown in Figure ane, nosotros tin create an expression that combines several variable terms, allowing us to solve this trouble and others similar it.

Figure

1

Showtime find the expanse of the square in square feet.

$$$$





A


=




s
2







=





(
ii
ten
)

2







=



4

10
two












A


=




s
2







=





(
2
10
)

ii







=



iv

x
2

Then discover the area of the triangle in square feet.

$$$$





A


=




1
2

b
h






=





1
2

(
two
x
)

(


three
ii


)







=





3
2

x











A


=




i
ii

b
h






=





ane
2

(
2
x
)

(


3
2


)







=





iii
2

x

Next discover the area of the rectangular door in square feet.

$$$$





A


=



50
westward






=



10
⋅
one






=


x










A


=



l
w






=



x
⋅
1






=


x

The expanse of the front end of the library can exist found by calculation the areas of the square and the triangle, and and so subtracting the surface area of the rectangle. When nosotros do this, we get

$$


4

10
2

+

3
2

ten
−
ten



ft

ii

,




4

x
2

+

3
2

10
−
x



ft

2

,

or

$$


4

ten
2

+

1
2

x




four

x
2

+

i
ii

x

ft^two.

In this section, we volition examine expressions such as this one, which combine several variable terms.

Identifying the Degree and Leading Coefficient of Polynomials

The formula just found is an example of a
polynomial, which is a sum of or departure of terms, each consisting of a variable raised to a nonnegative integer power. A number multiplied past a variable raised to an exponent, such every bit

$$


384
π
,




384
π
,

is known as a
coefficient. Coefficients can be positive, negative, or zero, and tin be whole numbers, decimals, or fractions. Each product

$$



a
i


x
i

,





a
i


10
i

,

such as

$$


384
π
due west
,




384
π
west
,

is a
term of a polynomial. If a term does not contain a variable, it is called a
constant.

A polynomial containing merely one term, such as

$$


5

x
iv

,




5

x
four

,

is chosen a
monomial. A polynomial containing two terms, such as

$$


two
10
−
9
,




ii
10
−
nine
,

is called a
binomial. A polynomial containing iii terms, such equally

$$


−3

x
2

+
8
x
−
seven
,




−3

10
2

+
8
ten
−
seven
,

is chosen a
trinomial.

We can find the
degree
of a polynomial by identifying the highest power of the variable that occurs in the polynomial. The term with the highest degree is called the
leading term
considering it is usually written first. The coefficient of the leading term is called the
leading coefficient. When a polynomial is written and then that the powers are descending, we say that it is in standard course.

Polynomials

A
polynomial
is an expression that can be written in the form

$$$$



a
n


10
due north

+
…
+

a
two


ten
two

+

a
i

x
+

a







a
northward


x
north

+
…
+

a
2


10
2

+

a
1

x
+

a

Each existent number
a_i
is called a
coefficient. The number

$$



a







a

that is not multiplied past a variable is called a
abiding. Each product

$$



a
i


10
i






a
i


x
i

is a
term of a polynomial. The highest power of the variable that occurs in the polynomial is called the
degree
of a polynomial. The
leading term
is the term with the highest power, and its coefficient is called the
leading coefficient.

How To

Given a polynomial expression, identify the degree and leading coefficient.

1. Observe the highest power of
   x
   to determine the degree.
2. Identify the term containing the highest ability of
   ten
   to find the leading term.
3. Identify the coefficient of the leading term.

Example

one

Identifying the Degree and Leading Coefficient of a Polynomial

For the following polynomials, place the caste, the leading term, and the leading coefficient.

1. ⓐ $$
   
   
   3
   +
   ii
   
   10
   2
   
   −
   4
   
   x
   3
   
   
   
   
   
   3
   +
   2
   
   ten
   two
   
   −
   4
   
   x
   3
   
   
   
2. ⓑ $$
   
   
   5
   
   t
   5
   
   −
   two
   
   t
   3
   
   +
   7
   t
   
   
   
   
   v
   
   t
   5
   
   −
   2
   
   t
   iii
   
   +
   seven
   t
   
   
3. ⓒ $$
   
   
   6
   p
   −
   
   p
   3
   
   −
   ii
   
   
   
   
   vi
   p
   −
   
   p
   3
   
   −
   two
   
   

Try It

#i

Identify the caste, leading term, and leading coefficient of the polynomial

$$


4

x
ii

−

x
6

+
2
x
−
6.




4

x
2

−

x
half dozen

+
2
x
−
vi.

Calculation and Subtracting Polynomials

Nosotros can add and decrease polynomials by combining like terms, which are terms that contain the same variables raised to the same exponents. For example,

$$


5

ten
2





5

ten
two

and

$$


−2

x
ii





−2

10
2

are like terms, and can be added to become

$$


three

x
2

,




3

x
2

,

but

$$


iii
x




three
x

and

$$


3

x
2





3

x
2

are not like terms, and therefore cannot be added.

How To

Given multiple polynomials, add or subtract them to simplify the expressions.

1. Combine like terms.
2. Simplify and write in standard form.

Instance

2

Calculation Polynomials

Find the sum.

$$



(

12

ten
2

+
9
10
−
21

)

+

(

four

x
3

+
8

x
2

−
5
10
+
20

)






(

12

x
ii

+
nine
x
−
21

)

+

(

iv

x
3

+
eight

x
two

−
5
10
+
xx

)

Analysis

We tin can check our answers to these types of problems using a graphing computer. To check, graph the problem as given forth with the simplified answer. The two graphs should exist equivalent. Exist sure to employ the aforementioned window to compare the graphs. Using different windows tin make the expressions seem equivalent when they are non.

Try It

#2

Find the sum.

$$



(

ii

x
three

+
5

ten
2

−
ten
+
i

)

+

(

2

10
2

−
iii
10
−
four

)






(

2

ten
3

+
5

x
2

−
10
+
1

)

+

(

2

x
2

−
iii
x
−
4

)

Example

3

Subtracting Polynomials

Notice the deviation.

$$



(

7

x
4

−

x
2

+
half-dozen
x
+
1

)

−

(

five

x
3

−
2

ten
2

+
3
ten
+
2

)






(

7

x
4

−

x
2

+
6
x
+
1

)

−

(

v

ten
3

−
2

x
2

+
3
x
+
2

)

Analysis

Notation that finding the departure between two polynomials is the same as adding the contrary of the 2nd polynomial to the first.

Try It

#3

Notice the difference.

$$



(

−7

10
3

−
7

x
2

+
six
x
−
2

)

−

(

iv

x
iii

−
6

x
2

−
x
+
vii

)






(

−7

ten
3

−
7

x
two

+
6
ten
−
2

)

−

(

4

x
iii

−
six

ten
two

−
x
+
7

)

Multiplying Polynomials

Multiplying polynomials is a scrap more challenging than adding and subtracting polynomials. Nosotros must apply the distributive property to multiply each term in the offset polynomial past each term in the second polynomial. We then combine like terms. We tin also utilise a shortcut called the
FOIL
method when multiplying binomials. Certain special products follow patterns that we can memorize and apply instead of multiplying the polynomials by hand each time. Nosotros will wait at a variety of means to multiply polynomials.

Multiplying Polynomials Using the Distributive Property

To multiply a number past a polynomial, we employ the distributive property. The number must exist distributed to each term of the polynomial. We can distribute the

$$


ii




two

in

$$


two
(
x
+
7
)




2
(
x
+
vii
)

to obtain the equivalent expression

$$


2
ten
+
fourteen.




two
x
+
14.

When multiplying polynomials, the distributive property allows us to multiply each term of the first polynomial past each term of the 2nd. We then add the products together and combine like terms to simplify.

How To

Given the multiplication of two polynomials, use the distributive property to simplify the expression.

1. Multiply each term of the start polynomial by each term of the 2d.
2. Combine like terms.
3. Simplify.

Example

4

Multiplying Polynomials Using the Distributive Property

Find the production.

$$



(

2
x
+
1

)


(

iii

x
ii

−
ten
+
4

)






(

2
x
+
1

)


(

3

ten
two

−
x
+
4

)

Analysis

Nosotros can use a tabular array to keep track of our work, as shown in Table ane. Write one polynomial across the top and the other down the side. For each box in the table, multiply the term for that row by the term for that cavalcade. Then add together all of the terms together, combine like terms, and simplify.

	$$ 3 x 2 3 x 2	$$ − x − ten	$$ + 4 + 4
$$ two 10 2 x	$$ 6 10 three 6 x iii	$$ −2 ten 2 −2 x 2	$$ eight 10 viii x
$$ + 1 + one	$$ 3 x two 3 x two	$$ − 10 − x	$$ four 4

Table

1

Try Information technology

#four

Notice the production.

$$


(
3
x
+
2
)

(


10
3

−
four

x
2

+
7

)





(
3
x
+
2
)

(


10
3

−
four

ten
2

+
vii

)

Using FOIL to Multiply Binomials

A shortcut called FOIL is sometimes used to discover the production of two binomials. Information technology is called FOIL because we multiply the
first terms, the
outer terms, the
inner terms, so the
last terms of each binomial.

The FOIL method arises out of the distributive property. Nosotros are but multiplying each term of the showtime binomial by each term of the second binomial, and and so combining similar terms.

How To

Given two binomials, use FOIL to simplify the expression.

1. Multiply the kickoff terms of each binomial.
2. Multiply the outer terms of the binomials.
3. Multiply the inner terms of the binomials.
4. Multiply the last terms of each binomial.
5. Add together the products.
6. Combine similar terms and simplify.

Case

5

Using FOIL to Multiply Binomials

Employ FOIL to find the product.

$$

(
2
x
–
xviii
)
(
3
10
+
3
)


(
ii
x
–
18
)
(
3
x
+
3
)

Try It

#5

Use FOIL to notice the product.

$$


(
x
+
7
)
(
3
ten
−
5
)




(
x
+
7
)
(
three
x
−
5
)

Perfect Foursquare Trinomials

Certain binomial products take special forms. When a binomial is squared, the result is called a
perfect square trinomial. We can find the square by multiplying the binomial by itself. However, at that place is a special class that each of these perfect square trinomials takes, and memorizing the grade makes squaring binomials much easier and faster. Let's look at a few perfect square trinomials to familiarize ourselves with the form.

$$$$










(
x
+
5
)

ii




=




x
two

+
10
x
+
25








(
x
−
3
)

2




=





x
ii

−
half dozen
x
+
ix








(
4
x
−
1
)

ii




=



16

x
2

−
8
ten
+
one
















(
x
+
5
)

2




=




10
two

+
10
ten
+
25








(
x
−
3
)

ii




=





x
two

−
6
ten
+
9








(
iv
ten
−
1
)

2




=



16

x
2

−
8
x
+
i

Discover that the first term of each trinomial is the foursquare of the first term of the binomial and, similarly, the last term of each trinomial is the square of the final term of the binomial. The middle term is double the product of the two terms. Lastly, we see that the offset sign of the trinomial is the same as the sign of the binomial.

Perfect Square Trinomials

When a binomial is squared, the result is the beginning term squared added to double the product of both terms and the last term squared.

$$$$




(
x
+
a
)

two

=
(
ten
+
a
)
(
10
+
a
)
=

10
two

+
2
a
10
+

a
2







(
10
+
a
)

ii

=
(
10
+
a
)
(
x
+
a
)
=

x
2

+
2
a
x
+

a
2

How To

Given a binomial, square it using the formula for perfect square trinomials.

1. Square the offset term of the binomial.
2. Square the last term of the binomial.
3. For the centre term of the trinomial, double the product of the two terms.
4. Add and simplify.

Example

half-dozen

Expanding Perfect Squares

Expand

$$




(
3
10
−
8
)

2

.






(
iii
10
−
8
)

2

.

Effort It

#six

Expand

$$




(
4
x
−
one
)

2

.






(
4
x
−
i
)

ii

.

Divergence of Squares

Another special product is called the
divergence of squares, which occurs when we multiply a binomial by another binomial with the same terms only the opposite sign. Let's see what happens when we multiply

$$


(
x
+
1
)
(
ten
−
one
)




(
x
+
i
)
(
x
−
1
)

using the FOIL method.

$$$$






(
x
+
1
)
(
10
−
1
)



=




x
2

−
ten
+
x
−
1






=




10
2

−
1












(
x
+
1
)
(
ten
−
ane
)



=




x
2

−
x
+
x
−
1






=




10
2

−
1

The centre term drops out, resulting in a difference of squares. Just as nosotros did with the perfect squares, allow'south look at a few examples.

$$$$






(
x
+
5
)
(
x
−
5
)



=




x
ii

−
25






(
x
+
11
)
(
x
−
11
)



=




x
two

−
121






(
two
10
+
3
)
(
2
x
−
3
)



=



iv

10
two

−
9












(
10
+
5
)
(
10
−
5
)



=




x
two

−
25






(
x
+
11
)
(
x
−
xi
)



=




ten
2

−
121






(
ii
ten
+
3
)
(
2
10
−
iii
)



=



iv

x
2

−
9

Because the sign changes in the 2d binomial, the outer and inner terms cancel each other out, and we are left only with the square of the first term minus the foursquare of the last term.

Q&A

Is there a special class for the sum of squares?

No. The difference of squares occurs considering the opposite signs of the binomials cause the middle terms to disappear. In that location are no two binomials that multiply to equal a sum of squares.

Divergence of Squares

When a binomial is multiplied by a binomial with the same terms separated by the reverse sign, the issue is the square of the commencement term minus the square of the concluding term.

$$$$


(
a
+
b
)
(
a
−
b
)
=

a
2

−

b
2





(
a
+
b
)
(
a
−
b
)
=

a
2

−

b
2

How To

Given a binomial multiplied past a binomial with the same terms just the opposite sign, find the difference of squares.

1. Square the first term of the binomials.
2. Foursquare the last term of the binomials.
3. Subtract the square of the last term from the foursquare of the first term.

Example

7

Multiplying Binomials Resulting in a Difference of Squares

Multiply

$$


(
9
ten
+
4
)
(
nine
ten
−
four
)
.




(
nine
x
+
4
)
(
9
ten
−
4
)
.

Attempt It

#seven

Multiply

$$


(
ii
x
+
7
)
(
2
x
−
vii
)
.




(
ii
x
+
7
)
(
2
10
−
7
)
.

Performing Operations with Polynomials of Several Variables

We have looked at polynomials containing only one variable. However, a polynomial tin can contain several variables. All of the same rules utilise when working with polynomials containing several variables. Consider an example:

$$$$






(
a
+
ii
b
)
(
4
a
−
b
−
c
)







a
(
four
a
−
b
−
c
)
+
2
b
(
4
a
−
b
−
c
)





Use the distributive holding
.






4

a
2

−
a
b
−
a
c
+
eight
a
b
−
2

b
2

−
2
b
c





Multiply
.






4

a
2

+
(
−
a
b
+
eight
a
b
)
−
a
c
−
2

b
2

−
ii
b
c





Combine like terms
.






iv

a
two

+
7
a
b
−
a
c
−
2
b
c
−
2

b
ii






Simplify
.












(
a
+
2
b
)
(
4
a
−
b
−
c
)







a
(
4
a
−
b
−
c
)
+
two
b
(
4
a
−
b
−
c
)





Use the distributive property
.






iv

a
2

−
a
b
−
a
c
+
viii
a
b
−
2

b
two

−
2
b
c





Multiply
.






iv

a
2

+
(
−
a
b
+
8
a
b
)
−
a
c
−
ii

b
two

−
2
b
c





Combine like terms
.






iv

a
two

+
7
a
b
−
a
c
−
2
b
c
−
2

b
2






Simplify
.

Example

8

Multiplying Polynomials Containing Several Variables

Multiply

$$


(
x
+
four
)
(
iii
x
−
2
y
+
5
)
.




(
10
+
4
)
(
3
ten
−
2
y
+
5
)
.

Effort It

#8

Multiply

$$


(
3
x
−
1
)
(
2
ten
+
vii
y
−
9
)
.




(
3
10
−
ane
)
(
2
10
+
seven
y
−
9
)
.

1.4 Section Exercises

Verbal

ane.

Evaluate the post-obit argument: The degree of a polynomial in standard class is the exponent of the leading term. Explain why the statement is true or false.

2
.

Many times, multiplying two binomials with two variables results in a trinomial. This is not the example when in that location is a divergence of two squares. Explicate why the product in this case is too a binomial.

3.

You can multiply polynomials with any number of terms and any number of variables using iv basic steps over and over until you reach the expanded polynomial. What are the 4 steps?

4
.

Land whether the following argument is true and explain why or why not: A trinomial is always a higher caste than a monomial.

Algebraic

For the following exercises, identify the degree of the polynomial.

5.

$$


7
x
−
2

x
two

+
13




7
10
−
2

x
2

+
13

vi
.

$$


xiv

m
3

+

g
2

−
16
m
+
eight




xiv

yard
3

+

chiliad
2

−
16
m
+
viii

seven.

$$


−625

a
8

+
xvi

b
iv





−625

a
8

+
sixteen

b
4

8
.

$$


200
p
−
xxx

p
2

m
+
40

m
3





200
p
−
30

p
ii

m
+
40

grand
iii

ten
.

$$


six

y
iv

−

y
v

+
3
y
−
four




vi

y
4

−

y
5

+
3
y
−
four

For the following exercises, find the sum or divergence.

11.

$$



(

12

10
2

+
3
x

)

−

(

8

10
2

−xix

)






(

12

x
two

+
3
10

)

−

(

8

x
ii

−xix

)

12
.

$$



(

4

z
iii

+
8

z
2

−
z

)

+

(

−2

z
two

+
z
+
half-dozen

)






(

4

z
3

+
eight

z
two

−
z

)

+

(

−2

z
2

+
z
+
6

)

13.

$$



(

6

w
two

+
24
w
+
24

)

−

(

3
westward

−


ii

half-dozen
w
+
iii

)






(

half dozen

w
2

+
24
w
+
24

)

−

(

three
w

−


2

6
due west
+
3

)

14
.

$$



(

7

a
3

+
6

a
2

−
iv
a
−
13

)

+

(

−
three

a
3

−
4

a
2

+
6
a
+
17

)






(

vii

a
iii

+
6

a
two

−
4
a
−
thirteen

)

+

(

−
3

a
three

−
4

a
2

+
6
a
+
17

)

fifteen.

$$



(

xi

b
iv

−
vi

b
3

+
xviii

b
two

−
four
b
+
8

)

−

(

3

b
3

+
6

b
2

+
3
b

)






(

11

b
four

−
6

b
3

+
xviii

b
2

−
4
b
+
8

)

−

(

three

b
3

+
6

b
2

+
three
b

)

16
.

$$



(

49

p
2

−
25

)

+

(

16

p
4

−
32

p
ii

+
16

)






(

49

p
ii

−
25

)

+

(

16

p
4

−
32

p
ii

+
16

)

For the following exercises, find the product.

17.

$$


(
4
10
+
two
)
(
6
x
−
4
)




(
iv
x
+
two
)
(
6
x
−
4
)

eighteen
.

$$



(

14

c
2

+
4
c

)


(

2

c
2

−
3
c

)






(

14

c
2

+
four
c

)


(

2

c
ii

−
3
c

)

19.

$$



(

vi

b
2

−
6

)


(

4

b
2

−
4

)






(

6

b
2

−
vi

)


(

4

b
two

−
4

)

20
.

$$


(
three
d
−
five
)
(
ii
d
+
9
)




(
3
d
−
5
)
(
ii
d
+
9
)

21.

$$


(
9
five
−
11
)
(
11
five
−
9
)




(
9
v
−
11
)
(
eleven
v
−
9
)

22
.

$$



(

iv

t
2

+
7
t

)


(

−3

t
2

+
four

)






(

4

t
ii

+
7
t

)


(

−iii

t
2

+
4

)

23.

$$


(
8
n
−
4
)

(


n
2

+
9

)





(
8
n
−
iv
)

(


northward
ii

+
9

)

For the following exercises, expand the binomial.

24
.

$$





(

4
x
+
5

)


ii








(

four
x
+
five

)


two

27.

$$





(

4
p
+
9

)


2








(

4
p
+
ix

)


2

xxx
.

$$





(

9
b
+
ane

)


2








(

9
b
+
1

)


2

For the following exercises, multiply the binomials.

31.

$$


(
4
c
+
1
)
(
4
c
−
ane
)




(
4
c
+
1
)
(
4
c
−
1
)

32
.

$$


(
9
a
−
4
)
(
9
a
+
four
)




(
nine
a
−
four
)
(
9
a
+
4
)

33.

$$


(
fifteen
north
−
6
)
(
15
n
+
6
)




(
15
n
−
6
)
(
15
n
+
6
)

34
.

$$


(
25
b
+
2
)
(
25
b
−
2
)




(
25
b
+
2
)
(
25
b
−
2
)

35.

$$


(
4
+
4
m
)
(
four
−
4
thou
)




(
iv
+
iv
g
)
(
4
−
4
m
)

36
.

$$


(
14
p
+
7
)
(
14
p
−
seven
)




(
fourteen
p
+
seven
)
(
xiv
p
−
7
)

37.

$$


(
11
q
−
10
)
(
11
q
+
10
)




(
11
q
−
10
)
(
eleven
q
+
ten
)

For the post-obit exercises, multiply the polynomials.

38
.

$$



(

2

x
ii

+
2
10
+
1

)

(
4
x
−
1
)





(

2

x
two

+
2
10
+
one

)

(
four
x
−
1
)

39.

$$



(

4

t
ii

+
t
−
vii

)


(

4

t
2

−
1

)






(

4

t
2

+
t
−
7

)


(

4

t
2

−
1

)

twoscore
.

$$


(
x
−
i
)

(


x
2

−
2
x
+
1

)





(
ten
−
1
)

(


x
2

−
2
ten
+
1

)

41.

$$


(
y
−
2
)

(


y
2

−
4
y
−
9

)





(
y
−
two
)

(


y
2

−
4
y
−
ix

)

42
.

$$


(
6
k
−
5
)

(

half dozen

grand
2

+
5
one thousand
−
1

)





(
6
k
−
5
)

(

6

thousand
2

+
five
g
−
1

)

43.

$$



(

three

p
2

+
2
p
−
10

)

(
p
−
1
)





(

3

p
two

+
2
p
−
ten

)

(
p
−
1
)

44
.

$$


(
four
m
−
thirteen
)

(

ii

thou
2

−
7
m
+
9

)





(
4
m
−
13
)

(

ii

grand
2

−
7
one thousand
+
9

)

45.

$$



(

a
+
b

)


(

a
−
b

)






(

a
+
b

)


(

a
−
b

)

46
.

$$


(
4
ten
−
6
y
)
(
half-dozen
x
−
4
y
)




(
4
10
−
6
y
)
(
6
x
−
4
y
)

48
.

$$


(
9
k
+
4
n
−
1
)
(
2
m
+
8
)




(
ix
m
+
4
northward
−
i
)
(
2
chiliad
+
8
)

49.

$$



(

4
t
−
x

)


(

t
−
10
+
ane

)






(

iv
t
−
ten

)


(

t
−
x
+
i

)

50
.

$$


(

b
two

−
one
)

(


a
2

+
ii
a
b
+

b
two


)





(

b
2

−
i
)

(


a
2

+
two
a
b
+

b
2


)

51.

$$



(

4
r
−
d

)


(

six
r
+
7
d

)






(

4
r
−
d

)


(

6
r
+
7
d

)

52
.

$$



(

x
+
y

)


(


x
2

−
ten
y
+

y
ii


)






(

ten
+
y

)


(


x
two

−
x
y
+

y
2


)

Existent-World Applications

53.

A developer wants to purchase a plot of land to build a business firm. The area of the plot tin can exist described by the following expression:

$$


(
iv
ten
+
1
)
(
viii
x
−
3
)




(
4
x
+
1
)
(
viii
10
−
3
)

where
10
is measured in meters. Multiply the binomials to find the area of the plot in standard class.

54
.

A prospective heir-apparent wants to know how much grain a specific silo tin can agree. The expanse of the floor of the silo is

$$





(

ii
10
+
9

)


2

.







(

two
10
+
9

)


2

.

The height of the silo is

$$


10
ten
+
10
,




10
x
+
x
,

where
x
is measured in feet. Expand the square and multiply by the height to find the expression that shows how much grain the silo can hold.

Extensions

For the following exercises, perform the given operations.

55.

$$




(
4
t
−
7
)

2

(
two
t
+
1
)
−

(

four

t
two

+
2
t
+
11

)







(
iv
t
−
7
)

2

(
2
t
+
1
)
−

(

4

t
2

+
2
t
+
xi

)

56
.

$$


(
three
b
+
6
)
(
3
b
−
6
)
(
9

b
two

−
36
)




(
iii
b
+
6
)
(
iii
b
−
6
)
(
ix

b
2

−
36
)

57.

$$


(

a
2

+
iv
a
c
+
four

c
2

)
(

a
2

−
4

c
ii

)




(

a
2

+
4
a
c
+
4

c
2

)
(

a
2

−
4

c
two

)

5-1 Practice Operations With Polynomials,

Source: https://en.asriportal.com/76339/5-1-4-practice-modeling-multiplying-polynomials/

Posted by: jacksonmallyn1966.blogspot.com

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