Contributed by:

This pdf includes the following topics:-

Polynomials and Factoring

monomial

binomial

trinomial

polynomial

degree of a monomial

Adding and Subtracting Polynomials

and many more.

Polynomials and Factoring

monomial

binomial

trinomial

polynomial

degree of a monomial

Adding and Subtracting Polynomials

and many more.

1.
Polynomials

Polynomials and

and

Factoring

Factoring

The basic building blocks of

algebraic expressions

Polynomials and

and

Factoring

Factoring

The basic building blocks of

algebraic expressions

2.
The height in feet of

a fireworks launched

straight up into the air

from (s) feet off the

ground at velocity (v) after

(t) seconds is given by the

equation:

-16t2 + vt + s

Find the height of a

firework launched from a 10

ft platform at 200 ft/s

after 5 seconds.

-16t2 + vt + s

-16(5)2 + 200(5) + 10

=400 + 1600 + 10

610 feet

a fireworks launched

straight up into the air

from (s) feet off the

ground at velocity (v) after

(t) seconds is given by the

equation:

-16t2 + vt + s

Find the height of a

firework launched from a 10

ft platform at 200 ft/s

after 5 seconds.

-16t2 + vt + s

-16(5)2 + 200(5) + 10

=400 + 1600 + 10

610 feet

3.
In regular math books, this is called

“substituting” or “evaluating”… We are given

the algebraic expression below and asked to

evaluate it.

x2 – 4x + 1

We need to find what this equals when we put a

number in for x.. Like

x=3

Everywhere you see an x… stick in a 3!

x2 – 4x + 1

= (3)2 – 4(3) + 1

= 9 – 12 + 1

= -2

“substituting” or “evaluating”… We are given

the algebraic expression below and asked to

evaluate it.

x2 – 4x + 1

We need to find what this equals when we put a

number in for x.. Like

x=3

Everywhere you see an x… stick in a 3!

x2 – 4x + 1

= (3)2 – 4(3) + 1

= 9 – 12 + 1

= -2

4.
What about x = -5?

Be careful with the negative! Use ( )!

x2 – 4x + 1

= (-5)2 – 4(-5) + 1

= 46

You try a couple

Use the same expression but let

x = 2 and

x = -1

Be careful with the negative! Use ( )!

x2 – 4x + 1

= (-5)2 – 4(-5) + 1

= 46

You try a couple

Use the same expression but let

x = 2 and

x = -1

5.
That critter in the last slide is a polynomial.

x2 – 4x + 1

Here are some others

x2 + 7x – 3

4a3 + 7a2 + a

nm2 – m

3x – 2

5

x2 – 4x + 1

Here are some others

x2 + 7x – 3

4a3 + 7a2 + a

nm2 – m

3x – 2

5

6.
For now (and, probably, forever) you

can just think of a polynomial as a

bunch to terms being added or

subtracted. The terms are just

products of numbers and letters

with exponents. As you’ll see later

on, polynomials have cool graphs.

can just think of a polynomial as a

bunch to terms being added or

subtracted. The terms are just

products of numbers and letters

with exponents. As you’ll see later

on, polynomials have cool graphs.

7.
Some math words to know!

monomial – is an expression that is a number, a

variable, or a product of a number and one or

more variables. Consequently, a monomial has no

variable in its denominator. It has one term.

(mono implies one).

13, 3x, -57, x2, 4y2, -2xy, or 520x2y2

(notice: no negative exponents, no fractional

exponents)

binomial – is the sum of two monomials. It has two

unlike terms (bi implies two).

3x + 1, x2 – 4x, 2x + y, or y – y2

monomial – is an expression that is a number, a

variable, or a product of a number and one or

more variables. Consequently, a monomial has no

variable in its denominator. It has one term.

(mono implies one).

13, 3x, -57, x2, 4y2, -2xy, or 520x2y2

(notice: no negative exponents, no fractional

exponents)

binomial – is the sum of two monomials. It has two

unlike terms (bi implies two).

3x + 1, x2 – 4x, 2x + y, or y – y2

8.
trinomial – is the sum of three monomials. It has

three unlike terms. (tri implies three).

x2 + 2x + 1, 3x2 – 4x + 10, 2x + 3y + 2

polynomial – is a monomial or the sum (+) or The ending of these

words “nomial” is Greek

difference (-) of one or more terms. for “part”.

(poly implies many).

x2 + 2x, 3x3 + x2 + 5x + 6, 4x + 6y + 8

• Polynomials are in simplest form when they contain no like

terms. x2 + 2x + 1 + 3x2 – 4x when simplified

becomes 4x2 – 2x + 1

• Polynomials are generally written in descending order.

Descending: 4x2 – 2x + 1 (exponents of variables decrease

from left to right)

Constants like 12 are monomials

since they can be written as 12x0 =

12 · 1 = 12 where the variable is x 0.

three unlike terms. (tri implies three).

x2 + 2x + 1, 3x2 – 4x + 10, 2x + 3y + 2

polynomial – is a monomial or the sum (+) or The ending of these

words “nomial” is Greek

difference (-) of one or more terms. for “part”.

(poly implies many).

x2 + 2x, 3x3 + x2 + 5x + 6, 4x + 6y + 8

• Polynomials are in simplest form when they contain no like

terms. x2 + 2x + 1 + 3x2 – 4x when simplified

becomes 4x2 – 2x + 1

• Polynomials are generally written in descending order.

Descending: 4x2 – 2x + 1 (exponents of variables decrease

from left to right)

Constants like 12 are monomials

since they can be written as 12x0 =

12 · 1 = 12 where the variable is x 0.

9.
The degree of a monomial - is the sum of the

exponents of its variables. For a nonzero

constant, the degree is 0. Zero has no degree.

Find the degree of each monomial

a) ¾x degree: 1 ¾x = ¾x1. The exponent is 1.

b) 7x2y3 degree: 5 The exponents are 2 and 3. Their sum is 5.

c) -4 degree: 0 The degree of a nonzero constant is 0.

exponents of its variables. For a nonzero

constant, the degree is 0. Zero has no degree.

Find the degree of each monomial

a) ¾x degree: 1 ¾x = ¾x1. The exponent is 1.

b) 7x2y3 degree: 5 The exponents are 2 and 3. Their sum is 5.

c) -4 degree: 0 The degree of a nonzero constant is 0.

10.
Here’s a polynomial

2x3 – 5x2 + x + 9

Each one of the little product things is a “term”.

2x3 – 5x2 + x + 9

term term term term

So, this guy has 4 terms.

2x3 – 5x2 + x + 9

The coefficients are the numbers in front of the letters.

2x3 - 5x2 + x + 9

NEXT

2 5 1 9

We just pretend

this last guy has a

Remember

letter behind him.

x=1·x

2x3 – 5x2 + x + 9

Each one of the little product things is a “term”.

2x3 – 5x2 + x + 9

term term term term

So, this guy has 4 terms.

2x3 – 5x2 + x + 9

The coefficients are the numbers in front of the letters.

2x3 - 5x2 + x + 9

NEXT

2 5 1 9

We just pretend

this last guy has a

Remember

letter behind him.

x=1·x

11.
Since “poly” means many, when there is only one term,

it’s a monomial:

5x

When there are two terms, it’s a binomial:

2x + 3

When there are three terms, it a trinomial:

x2 – x – 6

So, what about four terms? Quadnomial? Naw, we

won’t go there, too hard to pronounce.

This guy is just called a polynomial:

7x3 + 5x2 – 2x + 4 NEXT

it’s a monomial:

5x

When there are two terms, it’s a binomial:

2x + 3

When there are three terms, it a trinomial:

x2 – x – 6

So, what about four terms? Quadnomial? Naw, we

won’t go there, too hard to pronounce.

This guy is just called a polynomial:

7x3 + 5x2 – 2x + 4 NEXT

12.
So, there’s one word to remember to classify:

degree

Here’s how you find the degree of a polynomial:

Look at each term,

whoever has the most letters wins!

3x2 – 8x4 + x5

This guy has 5

letters…

The degree is 5.

This is a 7th degree polynomial:

6mn2 + m3n4 + 8

This guy has 7 letters…

The degree is 7 NEXT

degree

Here’s how you find the degree of a polynomial:

Look at each term,

whoever has the most letters wins!

3x2 – 8x4 + x5

This guy has 5

letters…

The degree is 5.

This is a 7th degree polynomial:

6mn2 + m3n4 + 8

This guy has 7 letters…

The degree is 7 NEXT

13.
This is a 1st degree polynomial

3x – 2

This guy has 1

letter…

The degree is 1.

By the way, the

coefficients don’t

have anything to

What about this dude?

This guy has no

do with the

degree.

8 letters…

The degree is 0.

How many letters does he have? ZERO!

So, he’s a zero degree polynomial

Before we go, I want you to know that Algebra

isn’t going to be just a bunch of weird words

that you don’t understand. I just needed to

start with some vocabulary so you’d know what

the heck we’re talking about!

3x – 2

This guy has 1

letter…

The degree is 1.

By the way, the

coefficients don’t

have anything to

What about this dude?

This guy has no

do with the

degree.

8 letters…

The degree is 0.

How many letters does he have? ZERO!

So, he’s a zero degree polynomial

Before we go, I want you to know that Algebra

isn’t going to be just a bunch of weird words

that you don’t understand. I just needed to

start with some vocabulary so you’d know what

the heck we’re talking about!

14.
3x4 + 5x2 – 7x + 1 term

term term term

The polynomial above is in standard form.

Standard form of a polynomial - means that

the degrees of its monomial terms decrease

from left to right.

Name using

Polynomial Degree Name using Number of number of

Degree Terms terms

7x + 4 1 Linear 2 Binomial

2

3x + 2x + 1 2 Quadratic 3 Trinomial

4x3 3 Cubic 1 Monomial

9x4 + 11x 4 Fourth degree 2 Binomial

5 0 Constant 1 monomial

Once you simplify a polynomial by

combining like terms, you can name the

polynomial based on degree or number of

monomials it contains.

term term term

The polynomial above is in standard form.

Standard form of a polynomial - means that

the degrees of its monomial terms decrease

from left to right.

Name using

Polynomial Degree Name using Number of number of

Degree Terms terms

7x + 4 1 Linear 2 Binomial

2

3x + 2x + 1 2 Quadratic 3 Trinomial

4x3 3 Cubic 1 Monomial

9x4 + 11x 4 Fourth degree 2 Binomial

5 0 Constant 1 monomial

Once you simplify a polynomial by

combining like terms, you can name the

polynomial based on degree or number of

monomials it contains.

15.
Classifying Polynomials

Write each polynomial in standard form. Then name each

polynomial based on its degree and the number of terms.

a) 5 – 2x

-2x + 5 Place terms in order.

linear binomial

b) 3x4 – 4 + 2x2 + 5x4 Place terms in order.

3x4 + 5x4 + 2x2 – 4 Combine like terms.

8x4 + 2x2 – 4

4th degree trinomial

Write each polynomial in standard form. Then name each

polynomial based on its degree and the number of terms.

a) 5 – 2x

-2x + 5 Place terms in order.

linear binomial

b) 3x4 – 4 + 2x2 + 5x4 Place terms in order.

3x4 + 5x4 + 2x2 – 4 Combine like terms.

8x4 + 2x2 – 4

4th degree trinomial

16.
Write each polynomial in standard

form. Then name each polynomial

based on its degree and the

number of terms.

a) 6x2 + 7 – 9x4

b) 3y – 4 – y3

c) 8 + 7v – 11v

form. Then name each polynomial

based on its degree and the

number of terms.

a) 6x2 + 7 – 9x4

b) 3y – 4 – y3

c) 8 + 7v – 11v

17.
Adding and

and Subtracting

Subtracting

Polynomials

Polynomials

The sum or difference

and Subtracting

Subtracting

Polynomials

Polynomials

The sum or difference

18.
Just as you can perform operations on

integers, you can perform operations on

polynomials. You can add polynomials using two

methods. Which one will you choose?

Closure of polynomials under addition or subtraction

The sum of two polynomials is a polynomial.

The difference of two polynomials is a polynomial.

integers, you can perform operations on

polynomials. You can add polynomials using two

methods. Which one will you choose?

Closure of polynomials under addition or subtraction

The sum of two polynomials is a polynomial.

The difference of two polynomials is a polynomial.

19.
Addition of

Polynomials You can rewrite each polynomial,

inserting a zero placeholder for

the “missing” term.

Method 1 (vertically)

Line up like terms. Then add the coefficients.

4x2 + 6x + 7 -2x3 + 2x2 – 5x + 3

2x2 – 9x + 1 0 + 5x2 + 4x - 5

6x2 – 3x + 8 -2x3 + 7x2 – x - 2

Method 2 (horizontally)

Group like terms. Then add the coefficients.

(4x2 + 6x + 7) + (2x2 – 9x + 1) = (4x2 + 2x2) + (6x – 9x) + (7 + 1)

= 6x2 – 3x + 8

Example 2:

(-2x3 + 0) + (2x2 + 5x2) + (-5x + 4x) + (3 – 5)

Example 2

Use a zero placeholder

Polynomials You can rewrite each polynomial,

inserting a zero placeholder for

the “missing” term.

Method 1 (vertically)

Line up like terms. Then add the coefficients.

4x2 + 6x + 7 -2x3 + 2x2 – 5x + 3

2x2 – 9x + 1 0 + 5x2 + 4x - 5

6x2 – 3x + 8 -2x3 + 7x2 – x - 2

Method 2 (horizontally)

Group like terms. Then add the coefficients.

(4x2 + 6x + 7) + (2x2 – 9x + 1) = (4x2 + 2x2) + (6x – 9x) + (7 + 1)

= 6x2 – 3x + 8

Example 2:

(-2x3 + 0) + (2x2 + 5x2) + (-5x + 4x) + (3 – 5)

Example 2

Use a zero placeholder

20.
Simplify each sum

• (12m2 + 4) + (8m2 + 5)

• (t2 – 6) + (3t2 + 11) Remember

Use a zero as a placeholder

for the “missing” term.

• (9w3 + 8w2) + (7w3 + 4)

• (2p3 + 6p2 + 10p) + (9p3 + 11p2 + 3p )

Word Problem

• (12m2 + 4) + (8m2 + 5)

• (t2 – 6) + (3t2 + 11) Remember

Use a zero as a placeholder

for the “missing” term.

• (9w3 + 8w2) + (7w3 + 4)

• (2p3 + 6p2 + 10p) + (9p3 + 11p2 + 3p )

Word Problem

21.
Find the perimeter of

each figure

9c - 10 - 6

17x

8x - 2

5c + 2

5x

9x

+

1

Recall that the

perimeter of a figure is

the sum of all the sides.

each figure

9c - 10 - 6

17x

8x - 2

5c + 2

5x

9x

+

1

Recall that the

perimeter of a figure is

the sum of all the sides.

22.
Subtracting

Polynomials

Earlier you learned that subtraction means to add the

opposite. So when you subtract a polynomial, change

the signs of each of the terms to its opposite. Then

add the coefficients.

Method 1 (vertically)

Line up like terms. Change the signs of the second polynomial,

then add. Simplify (2x3 + 5x2 – 3x) – (x3 – 8x2 + 11)

2x3 + 5x2 – 3x 2x3 + 5x2 – 3x

-(x3 – 8x2 + 0 + 11) -x3 + 8x2 + 0 - 11

x3 +13x2 – 3x - 11

Remember,

subtraction is adding

the opposite.

Method 2

Polynomials

Earlier you learned that subtraction means to add the

opposite. So when you subtract a polynomial, change

the signs of each of the terms to its opposite. Then

add the coefficients.

Method 1 (vertically)

Line up like terms. Change the signs of the second polynomial,

then add. Simplify (2x3 + 5x2 – 3x) – (x3 – 8x2 + 11)

2x3 + 5x2 – 3x 2x3 + 5x2 – 3x

-(x3 – 8x2 + 0 + 11) -x3 + 8x2 + 0 - 11

x3 +13x2 – 3x - 11

Remember,

subtraction is adding

the opposite.

Method 2

23.
Method 2 (horizontally)

Simplify (2x3 + 5x2 – 3x) – (x3 – 8x2 + 11)

Write the opposite of each term.

2x3 + 5x2 – 3x – x3 + 8x2 – 11

Group like terms.

(2x3 – x3) + (5x2 + 8x2) + (3x + 0) + (-11 + 0) =

x3 + 13x2 + 3x - 11 =

x3 + 13x2 + 3x - 11

Simplify (2x3 + 5x2 – 3x) – (x3 – 8x2 + 11)

Write the opposite of each term.

2x3 + 5x2 – 3x – x3 + 8x2 – 11

Group like terms.

(2x3 – x3) + (5x2 + 8x2) + (3x + 0) + (-11 + 0) =

x3 + 13x2 + 3x - 11 =

x3 + 13x2 + 3x - 11

24.

25.
Simplify each

subtraction

• (17n4 + 2n3) – (10n4 + n3)

• (24x5 + 12x) – (9x5 + 11x)

• 6c – 5 2b + 6 7h 2 + 4h - 8

-(4c + 9) -(b + 5) -(3h2 – 2h + 10)

subtraction

• (17n4 + 2n3) – (10n4 + n3)

• (24x5 + 12x) – (9x5 + 11x)

• 6c – 5 2b + 6 7h 2 + 4h - 8

-(4c + 9) -(b + 5) -(3h2 – 2h + 10)

26.
Multiplying and

and Factoring

Factoring

Using the Distributive Property

and Factoring

Factoring

Using the Distributive Property

27.
Observe the rectangle below. Remember that

the area A of a rectangle with length l and

width w is A = lw. So the area of this

rectangle is (4x)(2x), as shown.

4x

2x

A = lw

A = (4x)(2x)

****************************

x+x+x+x

x

+

x

The rectangle above shows the example that

4x = x + x + x + x and 2x = x + x NEXT

the area A of a rectangle with length l and

width w is A = lw. So the area of this

rectangle is (4x)(2x), as shown.

4x

2x

A = lw

A = (4x)(2x)

****************************

x+x+x+x

x

+

x

The rectangle above shows the example that

4x = x + x + x + x and 2x = x + x NEXT

28.
We can further divide the rectangle into

squares with side lengths of x.

x +x+x+ x

x

+

x

Since each side of the squares

are x, then x · x = x2

x+x+x+x

x x2 x2 x2 x2

+

x x2 x2 x2 x2

By applying the area formula

for a rectangle, the area of the

rectangle must be (4x)(2x).

This geometric model suggests the following

algebraic method for simplifying the product

of (4x)(2x).

(4x)(2x) = (4 · x)(2 · x) = (4 · 2)(x · x) = 8x 2

NEXT

Commutative Property Associative Property

squares with side lengths of x.

x +x+x+ x

x

+

x

Since each side of the squares

are x, then x · x = x2

x+x+x+x

x x2 x2 x2 x2

+

x x2 x2 x2 x2

By applying the area formula

for a rectangle, the area of the

rectangle must be (4x)(2x).

This geometric model suggests the following

algebraic method for simplifying the product

of (4x)(2x).

(4x)(2x) = (4 · x)(2 · x) = (4 · 2)(x · x) = 8x 2

NEXT

Commutative Property Associative Property

29.
To simplify a product of monomials

(4x)(2x)

• Use the Commutative and Associative Properties

of Multiplication to group the numerical

coefficients and to group like variable;

(4x)(2x) = (4 · 2)(x · x ) =

• Calculate the product of the numerical

coefficients; and

(4 · 2) = 8

• Use the properties of exponents to simplify the

variable product.

(x · x) = x1 · x1 = x1+1 = x2

Therefore (4x)(2x) = 8x2

(4x)(2x)

• Use the Commutative and Associative Properties

of Multiplication to group the numerical

coefficients and to group like variable;

(4x)(2x) = (4 · 2)(x · x ) =

• Calculate the product of the numerical

coefficients; and

(4 · 2) = 8

• Use the properties of exponents to simplify the

variable product.

(x · x) = x1 · x1 = x1+1 = x2

Therefore (4x)(2x) = 8x2

30.
You can also use the Distributive Property for

multiplying powers with the same base when

multiplying a polynomial by a monomial.

Simplify -4y2(5y4 – 3y2 + 2) Remember,

Multiply powers with the same base:

35 · 34 = 35 + 4 = 39

-4y2(5y4 – 3y2 + 2) =

-4y2(5y4) – 4y2(-3y2) – 4y2(2) = Use the Distributive Property

-20y2 + 4 + 12y2 + 2 – 8y2 = Multiply the coefficients and add the

-20y6 + 12y4 – 8y2 exponents of powers with the same base.

multiplying powers with the same base when

multiplying a polynomial by a monomial.

Simplify -4y2(5y4 – 3y2 + 2) Remember,

Multiply powers with the same base:

35 · 34 = 35 + 4 = 39

-4y2(5y4 – 3y2 + 2) =

-4y2(5y4) – 4y2(-3y2) – 4y2(2) = Use the Distributive Property

-20y2 + 4 + 12y2 + 2 – 8y2 = Multiply the coefficients and add the

-20y6 + 12y4 – 8y2 exponents of powers with the same base.

31.
Simplify each product.

a) 4b(5b2 + b + 6)

b) -7h(3h2 – 8h – 1)

c) 2x(x2 – 6x + 5)

Remember,

Multiplying powers with the same base.

d) 4y2(9y3 + 8y2 – 11)

a) 4b(5b2 + b + 6)

b) -7h(3h2 – 8h – 1)

c) 2x(x2 – 6x + 5)

Remember,

Multiplying powers with the same base.

d) 4y2(9y3 + 8y2 – 11)

32.
Factoring a Monomial

from a Polynomial Factoring a polynomial

reverses the

multiplication process.

Find the GCF of the terms of: To factor a monomial

4x3 + 12x2 – 8x from a polynomial, first

List the prime factors of each term. find the greatest

4x3 = 2 · 2 · x · x x common factor (GCF) of

12x2 = 2 · 2 · 3 · x · x its terms.

8x = 2 · 2 · 2 · x

The GCF is 2 · 2 · x or 4x.

from a Polynomial Factoring a polynomial

reverses the

multiplication process.

Find the GCF of the terms of: To factor a monomial

4x3 + 12x2 – 8x from a polynomial, first

List the prime factors of each term. find the greatest

4x3 = 2 · 2 · x · x x common factor (GCF) of

12x2 = 2 · 2 · 3 · x · x its terms.

8x = 2 · 2 · 2 · x

The GCF is 2 · 2 · x or 4x.

33.
Find the GCF of the terms of each polynomial.

a) 5v5 + 10v3

b) 3t2 – 18

c) 4b3 – 2b2 – 6b

d) 2x4 + 10x2 – 6x

a) 5v5 + 10v3

b) 3t2 – 18

c) 4b3 – 2b2 – 6b

d) 2x4 + 10x2 – 6x

34.
Factoring Out a To factor a polynomial

Monomial completely, you must factor

until there are no common

factors other than 1.

Factor 3x3 – 12x2 + 15x

Step 1

Find the GCF

3x3 = 3 · x · x · x Step 2

12x2 = 2 · 2 · 3 · x · x Factor out the GCF

15x = 3 · 5 · x 3x3 – 12x2 + 15x

The GCF is 3 · x or 3x = 3x(x2) + 3x(-4x) + 3x(5)

= 3x(x2 – 4x + 5)

Monomial completely, you must factor

until there are no common

factors other than 1.

Factor 3x3 – 12x2 + 15x

Step 1

Find the GCF

3x3 = 3 · x · x · x Step 2

12x2 = 2 · 2 · 3 · x · x Factor out the GCF

15x = 3 · 5 · x 3x3 – 12x2 + 15x

The GCF is 3 · x or 3x = 3x(x2) + 3x(-4x) + 3x(5)

= 3x(x2 – 4x + 5)

35.
Use the GCF to factor each polynomial.

a) 8x2 – 12x

b) 5d3 + 10d

c) 6m3 – 12m2 – 24m

d) 4x3 – 8x2 + 12x

Try to factor mentally by

scanning the coefficients of

each term to find the GCF.

Next, scan for the least power

of the variable.

a) 8x2 – 12x

b) 5d3 + 10d

c) 6m3 – 12m2 – 24m

d) 4x3 – 8x2 + 12x

Try to factor mentally by

scanning the coefficients of

each term to find the GCF.

Next, scan for the least power

of the variable.

36.
Multiplying Binomials

Binomials

Using the infamous FOIL method

Binomials

Using the infamous FOIL method

37.
Using the

Distributive

Property Distribute x + 4

As with the other Simplify: (2x + 3)(x + 4)

examples we have

seen, we can also (2x + 3)(x + 4) =

use the Distributive

Property to find the 2x(x + 4) + 3(x + 4) =

product of two 2x2 + 8x + 3x + 12 =

binomials. 2x2 + 11x + 12

Now Distribute 2x and 3

Distributive

Property Distribute x + 4

As with the other Simplify: (2x + 3)(x + 4)

examples we have

seen, we can also (2x + 3)(x + 4) =

use the Distributive

Property to find the 2x(x + 4) + 3(x + 4) =

product of two 2x2 + 8x + 3x + 12 =

binomials. 2x2 + 11x + 12

Now Distribute 2x and 3

38.
Simplify each

product.

a) (6h – 7)(2h + 3)

b) (5m + 2)(8m – 1)

c) (9a – 8)(7a + 4)

d) (2y – 3)(y + 2)

product.

a) (6h – 7)(2h + 3)

b) (5m + 2)(8m – 1)

c) (9a – 8)(7a + 4)

d) (2y – 3)(y + 2)

39.
Multiplying using FOIL

Another way to organize multiplying two binomials is

to use FOIL, which stands for,

“First, Outer, Inner, Last”. The term FOIL is a

memory device for applying the Distributive

Property to the product of two binomials.

Simplify (3x – 5)(2x + 7)

First Outer Inner Last

= (3x)(2x) + (3x)(7) – (5)(2x) – (5)(7)

(3x – 5)(2x + 7) = 6x2 + 21x - 10x - 35

= 6x2 + 11x - 35

The product is 6x2 + 11x - 35

Another way to organize multiplying two binomials is

to use FOIL, which stands for,

“First, Outer, Inner, Last”. The term FOIL is a

memory device for applying the Distributive

Property to the product of two binomials.

Simplify (3x – 5)(2x + 7)

First Outer Inner Last

= (3x)(2x) + (3x)(7) – (5)(2x) – (5)(7)

(3x – 5)(2x + 7) = 6x2 + 21x - 10x - 35

= 6x2 + 11x - 35

The product is 6x2 + 11x - 35

40.
Simplify each product

using FOIL

Remember,

First, Outer, Inner, Last

a) (3x + 4)(2x + 5)

b) (3x – 4)(2x + 5)

c) (3x + 4)(2x – 5)

d) (3x – 4)(2x – 5)

using FOIL

Remember,

First, Outer, Inner, Last

a) (3x + 4)(2x + 5)

b) (3x – 4)(2x + 5)

c) (3x + 4)(2x – 5)

d) (3x – 4)(2x – 5)

41.
Applying

Multiplication of area of outer rectangle =

Polynomials. (2x + 5)(3x + 1)

area of orange rectangle =

Find the area of the x(x + 2)

shaded (beige) region. area of shaded region

Simplify. = area of outer rectangle – area of

2x + 5 orange portion

(2x + 5)(3x + 1) – x(x + 2) =

x+2

6x2 + 15x + 2x + 5 – x2 – 2x =

3x + 1

x

6x2 – x2 + 15x + 2x – 2x + 5 =

5x2 + 17x + 5

Use the Distributive Property

Use the FOIL method to

to simplify –x(x + 2)

simplify (2x + 5)(3x + 1)

Multiplication of area of outer rectangle =

Polynomials. (2x + 5)(3x + 1)

area of orange rectangle =

Find the area of the x(x + 2)

shaded (beige) region. area of shaded region

Simplify. = area of outer rectangle – area of

2x + 5 orange portion

(2x + 5)(3x + 1) – x(x + 2) =

x+2

6x2 + 15x + 2x + 5 – x2 – 2x =

3x + 1

x

6x2 – x2 + 15x + 2x – 2x + 5 =

5x2 + 17x + 5

Use the Distributive Property

Use the FOIL method to

to simplify –x(x + 2)

simplify (2x + 5)(3x + 1)

42.
Find the area of the

shaded region.

Simplify.

Find the area of the green shaded region. Simplify.

5x + 8

6x + 2 5x

x+6

shaded region.

Simplify.

Find the area of the green shaded region. Simplify.

5x + 8

6x + 2 5x

x+6

43.
FOIL works when you are multiplying two binomials.

However, it does not work when multiplying a trinomial

and a binomial.

(You can use the vertical or horizontal method to distribute each term.)

Remember multiplying

Simplify (4x + x – 6)(2x – 3)

2 whole numbers.

312

Method 1 (vertical) x 23

936

4x2 + x - 6

624

7176

2x - 3

-12x2 - 3x + 18 Multiply by -3

8x3 + 2x2 - 12x Multiply by 2x

8x3 - 10x2 - 15x + 18 Add like terms

However, it does not work when multiplying a trinomial

and a binomial.

(You can use the vertical or horizontal method to distribute each term.)

Remember multiplying

Simplify (4x + x – 6)(2x – 3)

2 whole numbers.

312

Method 1 (vertical) x 23

936

4x2 + x - 6

624

7176

2x - 3

-12x2 - 3x + 18 Multiply by -3

8x3 + 2x2 - 12x Multiply by 2x

8x3 - 10x2 - 15x + 18 Add like terms

44.
Multiply using the

horizontal method.

Drawing arrows

Method 2 (horizontal) between terms can

help you identify all six

products.

(2x – 3)(4x2 + x – 6)

= 2x(4x2) + 2x(x) + 2x(-6) – 3(4x2) – 3(x) – 3(-6)

= 8x3 + 2x2 – 12x – 12x2 – 3x + 18

= 8x3 -10x2 - 15x + 18

The product is 8x3 – 10x2 – 15x + 18

horizontal method.

Drawing arrows

Method 2 (horizontal) between terms can

help you identify all six

products.

(2x – 3)(4x2 + x – 6)

= 2x(4x2) + 2x(x) + 2x(-6) – 3(4x2) – 3(x) – 3(-6)

= 8x3 + 2x2 – 12x – 12x2 – 3x + 18

= 8x3 -10x2 - 15x + 18

The product is 8x3 – 10x2 – 15x + 18

45.
Simplify using the Distributive Property.

a) (x + 2)(x + 5)

b) (2y + 5)(y – 3)

c) (h + 3)(h + 4)

Simplify using FOIL.

a) (r + 6)(r – 4)

b) (y + 4)(5y – 8)

c) (x – 7)(x + 9)

WORD PROBLEM

a) (x + 2)(x + 5)

b) (2y + 5)(y – 3)

c) (h + 3)(h + 4)

Simplify using FOIL.

a) (r + 6)(r – 4)

b) (y + 4)(5y – 8)

c) (x – 7)(x + 9)

WORD PROBLEM

46.
Find the area of the

green shaded region.

x+3

x+2

x-3 x

green shaded region.

x+3

x+2

x-3 x