The Least Common Multiple (LCM)

The Least Common Multiple (LCM)We say m is a multiple of x if m can be divided by x.

For example, 12 is a multiple of 2

The Least Common Multiple (LCM)We say m is a multiple of x if m can be divided by x.

For example, 12 is a multiple of 2. 12 is also a multiple of 3,

4, 6 or 12 itself.

The Least Common Multiple (LCM)We say m is a multiple of x if m can be divided by x.

For example, 12 is a multiple of 2. 12 is also a multiple of 3,

4, 6 or 12 itself.

The Least Common Multiple (LCM)

xy2 is a multiple of x,

We say m is a multiple of x if m can be divided by x.

For example, 12 is a multiple of 2. 12 is also a multiple of 3,

4, 6 or 12 itself.

The Least Common Multiple (LCM)

xy2 is a multiple of x, xy2 is also a multiple of y or xy.

We say m is a multiple of x if m can be divided by x.

For example, 12 is a multiple of 2. 12 is also a multiple of 3,

4, 6 or 12 itself.

The Least Common Multiple (LCM)

A common multiple of two or more quantities is a multiple of

all the given quantities.

xy2 is a multiple of x, xy2 is also a multiple of y or xy.

We say m is a multiple of x if m can be divided by x.

For example, 12 is a multiple of 2. 12 is also a multiple of 3,

4, 6 or 12 itself.

The Least Common Multiple (LCM)

A common multiple of two or more quantities is a multiple of

all the given quantities.

For example, 24 is a common multiple of 4 and 6,

xy2 is a multiple of x, xy2 is also a multiple of y or xy.

We say m is a multiple of x if m can be divided by x.

For example, 12 is a multiple of 2. 12 is also a multiple of 3,

4, 6 or 12 itself.

The Least Common Multiple (LCM)

A common multiple of two or more quantities is a multiple of

all the given quantities.

For example, 24 is a common multiple of 4 and 6,

xy2 is a multiple of x, xy2 is also a multiple of y or xy.

xy2 is a common multiple of x and y.

We say m is a multiple of x if m can be divided by x.

The least common multiple (LCM) of two or more numbers is

the smallest common multiple of all the given numbers.

For example, 12 is a multiple of 2. 12 is also a multiple of 3,

4, 6 or 12 itself.

The Least Common Multiple (LCM)

A common multiple of two or more quantities is a multiple of

all the given quantities.

For example, 24 is a common multiple of 4 and 6,

xy2 is a multiple of x, xy2 is also a multiple of y or xy.

xy2 is a common multiple of x and y.

We say m is a multiple of x if m can be divided by x.

The least common multiple (LCM) of two or more numbers is

the smallest common multiple of all the given numbers.

For example, 12 is a multiple of 2. 12 is also a multiple of 3,

4, 6 or 12 itself.

For example, 12 is the LCM of 4 and 6.

The Least Common Multiple (LCM)

A common multiple of two or more quantities is a multiple of

all the given quantities.

For example, 24 is a common multiple of 4 and 6,

xy2 is a multiple of x, xy2 is also a multiple of y or xy.

xy2 is a common multiple of x and y.

We say m is a multiple of x if m can be divided by x.

The least common multiple (LCM) of two or more numbers is

the smallest common multiple of all the given numbers.

For example, 12 is a multiple of 2. 12 is also a multiple of 3,

4, 6 or 12 itself.

For example, 12 is the LCM of 4 and 6,

xy is the LCM of x and y,

The Least Common Multiple (LCM)

A common multiple of two or more quantities is a multiple of

all the given quantities.

For example, 24 is a common multiple of 4 and 6,

xy2 is a multiple of x, xy2 is also a multiple of y or xy.

xy2 is a common multiple of x and y.

We say m is a multiple of x if m can be divided by x.

The least common multiple (LCM) of two or more numbers is

the smallest common multiple of all the given numbers.

For example, 12 is a multiple of 2. 12 is also a multiple of 3,

4, 6 or 12 itself.

For example, 12 is the LCM of 4 and 6,

xy is the LCM of x and y,

and that 12xy is the LCM of 4x and 6y.

The Least Common Multiple (LCM)

A common multiple of two or more quantities is a multiple of

all the given quantities.

For example, 24 is a common multiple of 4 and 6,

xy2 is a multiple of x, xy2 is also a multiple of y or xy.

xy2 is a common multiple of x and y.

We say m is a multiple of x if m can be divided by x.

The least common multiple (LCM) of two or more numbers is

the smallest common multiple of all the given numbers.

For example, 12 is a multiple of 2. 12 is also a multiple of 3,

4, 6 or 12 itself.

For example, 12 is the LCM of 4 and 6,

xy is the LCM of x and y,

and that 12xy is the LCM of 4x and 6y.

The Least Common Multiple (LCM)

A common multiple of two or more quantities is a multiple of

all the given quantities.

For example, 24 is a common multiple of 4 and 6,

We give two methods for finding the LCM.

I. The searching method

xy2 is a multiple of x, xy2 is also a multiple of y or xy.

xy2 is a common multiple of x and y.

We say m is a multiple of x if m can be divided by x.

The least common multiple (LCM) of two or more numbers is

the smallest common multiple of all the given numbers.

For example, 12 is a multiple of 2. 12 is also a multiple of 3,

4, 6 or 12 itself.

For example, 12 is the LCM of 4 and 6,

xy is the LCM of x and y,

and that 12xy is the LCM of 4x and 6y.

The Least Common Multiple (LCM)

A common multiple of two or more quantities is a multiple of

all the given quantities.

For example, 24 is a common multiple of 4 and 6,

We give two methods for finding the LCM.

I. The searching method

II. The construction method

xy2 is a multiple of x, xy2 is also a multiple of y or xy.

xy2 is a common multiple of x and y.

We say m is a multiple of x if m can be divided by x.

(LCM) The Searching MethodMethods of Finding LCM

The Minimum Coverage and The Construction Method

(LCM) The Searching Method

Example B. Find the LCM of 18, 24, 16.

Methods of Finding LCM

The Minimum Coverage and The Construction Method

(LCM) The Searching Method

To find the LCM of a list of numbers, test the multiples of

the largest number, one by one, until we find the LCM.

Example B. Find the LCM of 18, 24, 16.

Methods of Finding LCM

The Minimum Coverage and The Construction Method

(LCM) The Searching Method

To find the LCM of a list of numbers, test the multiples of

the largest number, one by one, until we find the LCM.

Example B. Find the LCM of 18, 24, 16.

List the multiples of 24:

Methods of Finding LCM

The Minimum Coverage and The Construction Method

The largest one on the list.

(LCM) The Searching Method

To find the LCM of a list of numbers, test the multiples of

the largest number, one by one, until we find the LCM.

Example B. Find the LCM of 18, 24, 16.

List the multiples of 24: 24, 48, 72, 96, 120, 144, …

Methods of Finding LCM

The Minimum Coverage and The Construction Method

The largest one on the list.

(LCM) The Searching Method

To find the LCM of a list of numbers, test the multiples of

the largest number, one by one, until we find the LCM.

Example B. Find the LCM of 18, 24, 16.

List the multiples of 24: 24, 48, 72, 96, 120, 144, …

Methods of Finding LCM

The Minimum Coverage and The Construction Method

The 1st one

that can also be

divided by 16&18.The largest one on the list.

(LCM) The Searching Method

To find the LCM of a list of numbers, test the multiples of

the largest number, one by one, until we find the LCM.

Example B. Find the LCM of 18, 24, 16.

List the multiples of 24: 24, 48, 72, 96, 120, 144, …

so the LCM of {18, 24, 16} is 144.

Methods of Finding LCM

The Minimum Coverage and The Construction Method

The 1st one

that can also be

divided by 16&18.

(LCM) The Searching Method

To find the LCM of a list of numbers, test the multiples of

the largest number, one by one, until we find the LCM.

Example B. Find the LCM of 18, 24, 16.

List the multiples of 24: 24, 48, 72, 96, 120, 144, …

so the LCM of {18, 24, 16} is 144.

Methods of Finding LCM

The Minimum Coverage and The Construction Method

The construction method is based on the principle of building

"the minimal (smallest, least) coverage which is necessary

to fulfill assorted requirements or conditions."

The 1st one

that can also be

divided by 16&18.

(LCM) The Searching Method

To find the LCM of a list of numbers, test the multiples of

the largest number, one by one, until we find the LCM.

Example B. Find the LCM of 18, 24, 16.

List the multiples of 24: 24, 48, 72, 96, 120, 144, …

so the LCM of {18, 24, 16} is 144.

Methods of Finding LCM

The Minimum Coverage and The Construction Method

The construction method is based on the principle of building

"the minimal (smallest, least) coverage which is necessary

to fulfill assorted requirements or conditions."

The 1st one

that can also be

divided by 16&18.

Following are two concrete examples that demonstrate

the principle of constructing the “the minimum coverage”.

(LCM) The Searching Method

To find the LCM of a list of numbers, test the multiples of

the largest number, one by one, until we find the LCM.

Example B. Find the LCM of 18, 24, 16.

List the multiples of 24: 24, 48, 72, 96, 120, 144, …

so the LCM of {18, 24, 16} is 144.

Methods of Finding LCM

The Minimum Coverage and The Construction Method

The construction method is based on the principle of building

"the minimal (smallest, least) coverage which is necessary

to fulfill assorted requirements or conditions."

Following are two concrete examples that demonstrate

the principle of constructing the “the minimum coverage”.

We will construct the LCD utilizing this principle later.

The 1st one

that can also be

divided by 16&18.

The minimal-coverage-construction method takes just

enough of each specification to build the minimum which

"covers" all the requirements on the list.

Methods of Finding LCM

The minimal-coverage-construction method takes just

enough of each specification to build the minimum which

"covers" all the requirements on the list. The following is an

example that illustrates the principle behind this method .

Methods of Finding LCM

Methods of Finding LCMThe minimal-coverage-construction method takes just

enough of each specification to build the minimum which

"covers" all the requirements on the list. The following is an

example that illustrates the principle behind this method .

Example B. A student wants to take enough courses so she

meets the requirement to apply to colleges A, B, and C.

The following is a table of requirements for each college,

how many years of each subject does she need?

Science Math. English Arts

A 2 yrs 3 yrs 3 yrs

B 3 yrs 3 yrs 2 yrs

C 2 yrs 2 yrs 4 yrs 1 yrs

Methods of Finding LCMThe minimal-coverage-construction method takes just

enough of each specification to build the minimum which

"covers" all the requirements on the list. The following is an

example that illustrates the principle behind this method .

Example B. A student wants to take enough courses so she

meets the requirement to apply to colleges A, B, and C.

The following is a table of requirements for each college,

how many years of each subject does she need?

Science Math. English Arts

A 2 yrs 3 yrs 3 yrs

B 3 yrs 3 yrs 2 yrs

C 2 yrs 2 yrs 4 yrs 1 yrs

Science-3 yr

Methods of Finding LCMThe minimal-coverage-construction method takes just

enough of each specification to build the minimum which

"covers" all the requirements on the list. The following is an

example that illustrates the principle behind this method .

Example B. A student wants to take enough courses so she

meets the requirement to apply to colleges A, B, and C.

The following is a table of requirements for each college,

how many years of each subject does she need?

Science Math. English Arts

A 2 yrs 3 yrs 3 yrs

B 3 yrs 3 yrs 2 yrs

C 2 yrs 2 yrs 4 yrs 1 yrs

Science-3 yr Math-3yrs

Methods of Finding LCMThe minimal-coverage-construction method takes just

enough of each specification to build the minimum which

"covers" all the requirements on the list. The following is an

example that illustrates the principle behind this method .

Example B. A student wants to take enough courses so she

meets the requirement to apply to colleges A, B, and C.

The following is a table of requirements for each college,

how many years of each subject does she need?

Science Math. English Arts

A 2 yrs 3 yrs 3 yrs

B 3 yrs 3 yrs 2 yrs

C 2 yrs 2 yrs 4 yrs 1 yrs

Science-3 yr Math-3yrs English-4 yrs Arts-1yr

Methods of Finding LCMThe minimal-coverage-construction method takes just

enough of each specification to build the minimum which

"covers" all the requirements on the list. The following is an

example that illustrates the principle behind this method .

Example B. A student wants to take enough courses so she

meets the requirement to apply to colleges A, B, and C.

The following is a table of requirements for each college,

how many years of each subject does she need?

Science Math. English Arts

A 2 yrs 3 yrs 3 yrs

B 3 yrs 3 yrs 2 yrs

C 2 yrs 2 yrs 4 yrs 1 yrs

Science-3 yr Math-3yrs English-4 yrs Arts-1yr

Methods of Finding LCMThe minimal-coverage-construction method takes just

enough of each specification to build the minimum which

"covers" all the requirements on the list. The following is an

example that illustrates the principle behind this method .

Example B. A student wants to take enough courses so she

meets the requirement to apply to colleges A, B, and C.

The following is a table of requirements for each college,

how many years of each subject does she need?

form the least amount of required classes needed to be taken.

LCM and LCDHere is another example of fulfilling "the minimal" requirements.

Example C. There are three identical

boxes with the same content.

Apu took some items from the 1st

box and what’s left is shown here.

LCM and LCDHere is another example of fulfilling "the minimal" requirements.

Example C. There are three identical

boxes with the same content.

Apu took some items from the 1st

box and what’s left is shown here.

LCM and LCD

What’s left after

Apu took some

items.

Here is another example of fulfilling "the minimal" requirements.

Example C. There are three identical

boxes with the same content.

Apu took some items from the 1st

box and what’s left is shown here.

Bolo took some items from the 2nd

box and what’s left is shown here.

LCM and LCD

What’s left after

Apu took some

items.

What’s left after

Bolo took some

items.

Here is another example of fulfilling "the minimal" requirements.

Example C. There are three identical

boxes with the same content.

Apu took some items from the 1st

box and what’s left is shown here.

Bolo took some items from the 2nd

box and what’s left is shown here.

Cato took some items from the 3rd

box and what’s left is shown here.

LCM and LCD

What’s left after

Apu took some

items.

What’s left after

Bolo took some

items.

What’s left after

Cato took some

items.

Here is another example of fulfilling "the minimal" requirements.

Example C. There are three identical

boxes with the same content.

Apu took some items from the 1st

box and what’s left is shown here.

Bolo took some items from the 2nd

box and what’s left is shown here.

Cato took some items from the 3rd

box and what’s left is shown here.

What is the least amount of items

possible in box originaIly?

What’s the least amount of items

that each person took?

LCM and LCD

What’s left after

Apu took some

items.

What’s left after

Bolo took some

items.

What’s left after

Cato took some

items.

Here is another example of fulfilling "the minimal" requirements.

Example C. There are three identical

boxes with the same content.

Apu took some items from the 1st

box and what’s left is shown here.

Bolo took some items from the 2nd

box and what’s left is shown here.

Cato took some items from the 3rd

box and what’s left is shown here.

What is the least amount of items

possible in box originaIly?

What’s the least amount of items

that each person took?

LCM and LCD

What’s left after

Apu took some

items.

What’s left after

Bolo took some

items.

What’s left after

Cato took some

items.

The least amount of items in the box consist of 2 apples,

Here is another example of fulfilling "the minimal" requirements.

Example C. There are three identical

boxes with the same content.

Apu took some items from the 1st

box and what’s left is shown here.

Bolo took some items from the 2nd

box and what’s left is shown here.

Cato took some items from the 3rd

box and what’s left is shown here.

What is the least amount of items

possible in box originaIly?

What’s the least amount of items

that each person took?

LCM and LCD

What’s left after

Apu took some

items.

What’s left after

Bolo took some

items.

What’s left after

Cato took some

items.

The least amount of items in the box consist of 2 apples,

5 bananas and 4 carrots.

Here is another example of fulfilling "the minimal" requirements.

Example C. There are three identical

boxes with the same content.

Apu took some items from the 1st

box and what’s left is shown here.

Bolo took some items from the 2nd

box and what’s left is shown here.

Cato took some items from the 3rd

box and what’s left is shown here.

What is the least amount of items

possible in box originaIly?

What’s the least amount of items

that each person took?

LCM and LCD

What’s left after

Apu took some

items.

What’s left after

Bolo took some

items.

What’s left after

Cato took some

items.

The least amount of items in the box consist of 2 apples,

5 bananas and 4 carrots. Apu took 1 apple and 1 banana,

Here is another example of fulfilling "the minimal" requirements.

Example C. There are three identical

boxes with the same content.

Apu took some items from the 1st

box and what’s left is shown here.

Bolo took some items from the 2nd

box and what’s left is shown here.

Cato took some items from the 3rd

box and what’s left is shown here.

What is the least amount of items

possible in box originaIly?

What’s the least amount of items

that each person took?

LCM and LCD

What’s left after

Apu took some

items.

What’s left after

Bolo took some

items.

What’s left after

Cato took some

items.

The least amount of items in the box consist of 2 apples,

5 bananas and 4 carrots. Apu took 1 apple and 1 banana,

Bolo took 2 carrots, and Cato took 2 banana and 1 carrot.

Here is another example of fulfilling "the minimal" requirements.

But when the LCM is large, the listing method is cumbersome.

LCM and LCD

But when the LCM is large, the listing method is cumbersome.

It's easier to find the LCM by constructing it instead because

the LCM is a type of minimal coverage of the list of numbers.

LCM and LCD

Example D. Construct the LCM of {8, 15, 18}.

But when the LCM is large, the listing method is cumbersome.

It's easier to find the LCM by constructing it instead because

the LCM is a type of minimal coverage of the list of numbers.

LCM and LCD

The LCM–Construction Method

To construct the LCM:

a. Factor each number completely

Example D. Construct the LCM of {8, 15, 18}.

But when the LCM is large, the listing method is cumbersome.

It's easier to find the LCM by constructing it instead because

the LCM is a type of minimal coverage of the list of numbers.

LCM and LCD

The LCM–Construction Method

To construct the LCM:

a. Factor each number completely

Example D. Construct the LCM of {8, 15, 18}.

Factor each number completely,

But when the LCM is large, the listing method is cumbersome.

It's easier to find the LCM by constructing it instead because

the LCM is a type of minimal coverage of the list of numbers.

LCM and LCD

The LCM–Construction Method

To construct the LCM:

a. Factor each number completely

Example D. Construct the LCM of {8, 15, 18}.

Factor each number completely,

8 = 23

15 = 3 * 5

18 = 2 * 32

But when the LCM is large, the listing method is cumbersome.

It's easier to find the LCM by constructing it instead because

the LCM is a type of minimal coverage of the list of numbers.

LCM and LCD

The LCM–Construction Method

To construct the LCM:

a. Factor each number completely

b. For each prime factor, take the highest power appearing in

the factorizations. The LCM is their product.

Example D. Construct the LCM of {8, 15, 18}.

Factor each number completely,

8 = 23

15 = 3 * 5

18 = 2 * 32

But when the LCM is large, the listing method is cumbersome.

It's easier to find the LCM by constructing it instead because

the LCM is a type of minimal coverage of the list of numbers.

LCM and LCD

The LCM–Construction Method

To construct the LCM:

a. Factor each number completely

b. For each prime factor, take the highest power appearing in

the factorizations. The LCM is their product.

Example D. Construct the LCM of {8, 15, 18}.

Factor each number completely,

8 = 23

15 = 3 * 5

18 = 2 * 32

From the factorization select the highest degree of each prime

factor:

But when the LCM is large, the listing method is cumbersome.

It's easier to find the LCM by constructing it instead because

the LCM is a type of minimal coverage of the list of numbers.

LCM and LCD

The LCM–Construction Method

To construct the LCM:

a. Factor each number completely

b. For each prime factor, take the highest power appearing in

the factorizations. The LCM is their product.

Example D. Construct the LCM of {8, 15, 18}.

Factor each number completely,

8 = 23

15 = 3 * 5

18 = 2 * 32

From the factorization select the highest degree of each prime

factor: 23,

But when the LCM is large, the listing method is cumbersome.

It's easier to find the LCM by constructing it instead because

the LCM is a type of minimal coverage of the list of numbers.

LCM and LCD

The LCM–Construction Method

The LCM must be divisible by 8 = 23

so it must have the factor 23

To construct the LCM:

a. Factor each number completely

b. For each prime factor, take the highest power appearing in

the factorizations. The LCM is their product.

Example D. Construct the LCM of {8, 15, 18}.

Factor each number completely,

8 = 23

15 = 3 * 5

18 = 2 * 32

From the factorization select the highest degree of each prime

factor: 23, 32

But when the LCM is large, the listing method is cumbersome.

It's easier to find the LCM by constructing it instead because

the LCM is a type of minimal coverage of the list of numbers.

LCM and LCD

The LCM–Construction Method

The LCM must be divisible by 18 = 2*32

so it must have the factor 32

(we have enough 2’s already)

To construct the LCM:

a. Factor each number completely

b. For each prime factor, take the highest power appearing in

the factorizations. The LCM is their product.

Example D. Construct the LCM of {8, 15, 18}.

Factor each number completely,

8 = 23

15 = 3 * 5

18 = 2 * 32

From the factorization select the highest degree of each prime

factor: 23, 32, 5,

But when the LCM is large, the listing method is cumbersome.

It's easier to find the LCM by constructing it instead because

the LCM is a type of minimal coverage of the list of numbers.

LCM and LCD

The LCM–Construction Method

The LCM must has the factor 5

to be divisible by 15 = 3*5

(we have enough 3’s already)

To construct the LCM:

a. Factor each number completely

b. For each prime factor, take the highest power appearing in

the factorizations. The LCM is their product.

Example D. Construct the LCM of {8, 15, 18}.

Factor each number completely,

8 = 23

15 = 3 * 5

18 = 2 * 32

From the factorization select the highest degree of each prime

factor: 23, 32, 5, so the LCM{8, 15, 18} = 23*32

*5 = 8*9*5 = 360.

But when the LCM is large, the listing method is cumbersome.

It's easier to find the LCM by constructing it instead because

the LCM is a type of minimal coverage of the list of numbers.

LCM and LCD

The LCM–Construction Method

b. Construct the LCM of x2y3z, x3yz4, x3zw

Methods of Finding LCM

b. Construct the LCM of x2y3z, x3yz4, x3zw

All quantities are factored, taking the highest power of each

factor. The LCM =

Methods of Finding LCM

b. Construct the LCM of x2y3z, x3yz4, x3zw

All quantities are factored, taking the highest power of each

factor. The LCM = x3

Methods of Finding LCM

b. Construct the LCM of x2y3z, x3yz4, x3zw

All quantities are factored, taking the highest power of each

factor. The LCM = x3y3

Methods of Finding LCM

b. Construct the LCM of x2y3z, x3yz4, x3zw

All quantities are factored, taking the highest power of each

factor. The LCM = x3y3z4

Methods of Finding LCM

b. Construct the LCM of x2y3z, x3yz4, x3zw

All quantities are factored, taking the highest power of each

factor. The LCM = x3y3z4w.

Methods of Finding LCM

b. Construct the LCM of x2y3z, x3yz4, x3zw

All quantities are factored, taking the highest power of each

factor. The LCM = x3y3z4w.

Methods of Finding LCM

c. Construct the LCM of x2 – 3x + 2, x2 – x – 2

b. Construct the LCM of x2y3z, x3yz4, x3zw

All quantities are factored, taking the highest power of each

factor. The LCM = x3y3z4w.

Methods of Finding LCM

c. Construct the LCM of x2 – 3x + 2, x2 – x – 2

Factor each quantity.

b. Construct the LCM of x2y3z, x3yz4, x3zw

All quantities are factored, taking the highest power of each

factor. The LCM = x3y3z4w.

Methods of Finding LCM

c. Construct the LCM of x2 – 3x + 2, x2 + x – 2

Factor each quantity.

x2 – 3x + 2 = (x – 2)(x – 1)

b. Construct the LCM of x2y3z, x3yz4, x3zw

All quantities are factored, taking the highest power of each

factor. The LCM = x3y3z4w.

Methods of Finding LCM

c. Construct the LCM of x2 – 3x + 2, x2 + x – 2

Factor each quantity.

x2 – 3x + 2 = (x – 2)(x – 1)

x2 + x – 2 = (x + 2)(x – 1)

b. Construct the LCM of x2y3z, x3yz4, x3zw

All quantities are factored, taking the highest power of each

factor. The LCM = x3y3z4w.

Methods of Finding LCM

c. Construct the LCM of x2 – 3x + 2, x2 + x – 2

Factor each quantity.

x2 – 3x + 2 = (x – 2)(x – 1)

x2 + x – 2 = (x + 2)(x – 1)

Taking the highest power of each factor to get the

LCM =

b. Construct the LCM of x2y3z, x3yz4, x3zw

All quantities are factored, taking the highest power of each

factor. The LCM = x3y3z4w.

Methods of Finding LCM

c. Construct the LCM of x2 – 3x + 2, x2 + x – 2

Factor each quantity.

x2 – 3x + 2 = (x – 2)(x – 1)

x2 + x – 2 = (x + 2)(x – 1)

Taking the highest power of each factor to get the

LCM = (x – 2)

b. Construct the LCM of x2y3z, x3yz4, x3zw

All quantities are factored, taking the highest power of each

factor. The LCM = x3y3z4w.

Methods of Finding LCM

c. Construct the LCM of x2 – 3x + 2, x2 + x – 2

Factor each quantity.

x2 – 3x + 2 = (x – 2)(x – 1)

x2 + x – 2 = (x + 2)(x – 1)

Taking the highest power of each factor to get the

LCM = (x – 2)(x – 1)

b. Construct the LCM of x2y3z, x3yz4, x3zw

All quantities are factored, taking the highest power of each

factor. The LCM = x3y3z4w.

Methods of Finding LCM

c. Construct the LCM of x2 – 3x + 2, x2 + x – 2

Factor each quantity.

x2 – 3x + 2 = (x – 2)(x – 1)

x2 + x – 2 = (x + 2)(x – 1)

Taking the highest power of each factor to get the

LCM = (x – 2)(x – 1)(x +2)

b. Construct the LCM of x2y3z, x3yz4, x3zw

All quantities are factored, taking the highest power of each

factor. The LCM = x3y3z4w.

Methods of Finding LCM

d. Construct the LCM of x4 – 4x2, x2 – 4x +4

c. Construct the LCM of x2 – 3x + 2, x2 + x – 2

Factor each quantity.

x2 – 3x + 2 = (x – 2)(x – 1)

x2 + x – 2 = (x + 2)(x – 1)

Taking the highest power of each factor to get the

LCM = (x – 2)(x – 1)(x +2)

b. Construct the LCM of x2y3z, x3yz4, x3zw

All quantities are factored, taking the highest power of each

factor. The LCM = x3y3z4w.

Methods of Finding LCM

d. Construct the LCM of x4 – 4x2, x2 – 4x +4

Factor each quantity.

c. Construct the LCM of x2 – 3x + 2, x2 + x – 2

Factor each quantity.

x2 – 3x + 2 = (x – 2)(x – 1)

x2 + x – 2 = (x + 2)(x – 1)

Taking the highest power of each factor to get the

LCM = (x – 2)(x – 1)(x +2)

b. Construct the LCM of x2y3z, x3yz4, x3zw

All quantities are factored, taking the highest power of each

factor. The LCM = x3y3z4w.

Methods of Finding LCM

d. Construct the LCM of x4 – 4x2, x2 – 4x +4

Factor each quantity.

x4 – 4x2 = x2(x2 – 4)

c. Construct the LCM of x2 – 3x + 2, x2 + x – 2

Factor each quantity.

x2 – 3x + 2 = (x – 2)(x – 1)

x2 + x – 2 = (x + 2)(x – 1)

Taking the highest power of each factor to get the

LCM = (x – 2)(x – 1)(x +2)

b. Construct the LCM of x2y3z, x3yz4, x3zw

All quantities are factored, taking the highest power of each

factor. The LCM = x3y3z4w.

Methods of Finding LCM

d. Construct the LCM of x4 – 4x2, x2 – 4x +4

Factor each quantity.

x4 – 4x2 = x2(x2 – 4) = x2(x + 2)(x – 2)

c. Construct the LCM of x2 – 3x + 2, x2 + x – 2

Factor each quantity.

x2 – 3x + 2 = (x – 2)(x – 1)

x2 + x – 2 = (x + 2)(x – 1)

Taking the highest power of each factor to get the

LCM = (x – 2)(x – 1)(x +2)

b. Construct the LCM of x2y3z, x3yz4, x3zw

All quantities are factored, taking the highest power of each

factor. The LCM = x3y3z4w.

Methods of Finding LCM

d. Construct the LCM of x4 – 4x2, x2 – 4x +4

Factor each quantity.

x4 – 4x2 = x2(x2 – 4) = x2(x + 2)(x – 2)

x2 – 4x + 4 = (x – 2)2

c. Construct the LCM of x2 – 3x + 2, x2 + x – 2

Factor each quantity.

x2 – 3x + 2 = (x – 2)(x – 1)

x2 + x – 2 = (x + 2)(x – 1)

Taking the highest power of each factor to get the

LCM = (x – 2)(x – 1)(x +2)

b. Construct the LCM of x2y3z, x3yz4, x3zw

All quantities are factored, taking the highest power of each

factor. The LCM = x3y3z4w.

Methods of Finding LCM

d. Construct the LCM of x4 – 4x2, x2 – 4x +4

Factor each quantity.

x4 – 4x2 = x2(x2 – 4) = x2(x + 2)(x – 2)

x2 – 4x + 4 = (x – 2)2

Taking the highest power of each factor to get the

LCM =

c. Construct the LCM of x2 – 3x + 2, x2 + x – 2

Factor each quantity.

x2 – 3x + 2 = (x – 2)(x – 1)

x2 + x – 2 = (x + 2)(x – 1)

Taking the highest power of each factor to get the

LCM = (x – 2)(x – 1)(x +2)

b. Construct the LCM of x2y3z, x3yz4, x3zw

All quantities are factored, taking the highest power of each

factor. The LCM = x3y3z4w.

Methods of Finding LCM

d. Construct the LCM of x4 – 4x2, x2 – 4x +4

Factor each quantity.

x4 – 4x2 = x2(x2 – 4) = x2(x + 2)(x – 2)

x2 – 4x + 4 = (x – 2)2

Taking the highest power of each factor to get the

LCM = x2

c. Construct the LCM of x2 – 3x + 2, x2 + x – 2

Factor each quantity.

x2 – 3x + 2 = (x – 2)(x – 1)

x2 + x – 2 = (x + 2)(x – 1)

Taking the highest power of each factor to get the

LCM = (x – 2)(x – 1)(x +2)

b. Construct the LCM of x2y3z, x3yz4, x3zw

All quantities are factored, taking the highest power of each

factor. The LCM = x3y3z4w.

Methods of Finding LCM

d. Construct the LCM of x4 – 4x2, x2 – 4x +4

Factor each quantity.

x4 – 4x2 = x2(x2 – 4) = x2(x + 2)(x – 2)

x2 – 4x + 4 = (x – 2)2

Taking the highest power of each factor to get the

LCM = x2(x + 2)

c. Construct the LCM of x2 – 3x + 2, x2 + x – 2

Factor each quantity.

x2 – 3x + 2 = (x – 2)(x – 1)

x2 + x – 2 = (x + 2)(x – 1)

Taking the highest power of each factor to get the

LCM = (x – 2)(x – 1)(x +2)

b. Construct the LCM of x2y3z, x3yz4, x3zw

All quantities are factored, taking the highest power of each

factor. The LCM = x3y3z4w.

Methods of Finding LCM

d. Construct the LCM of x4 – 4x2, x2 – 4x +4

Factor each quantity.

x4 – 4x2 = x2(x2 – 4) = x2(x + 2)(x – 2)

x2 – 4x + 4 = (x – 2)2

Taking the highest power of each factor to get the

LCM = x2(x + 2)(x – 2)2

c. Construct the LCM of x2 – 3x + 2, x2 + x – 2

Factor each quantity.

x2 – 3x + 2 = (x – 2)(x – 1)

x2 + x – 2 = (x + 2)(x – 1)

Taking the highest power of each factor to get the

LCM = (x – 2)(x – 1)(x +2)

Clearing Denominator with LCD

The LCD is used for clearing denominators to simplify fractional calculations.

Clearing Denominator with LCD

Clearing Denominator with LCDThe LCD is used for clearing denominators to simplify fractional calculations. Likewise the LCD is utilized in the same manner and purposes for simplifying rational expression calculations.

Clearing Denominator with LCDThe LCD is used for clearing denominators to simplify fractional calculations. Likewise the LCD is utilized in the same manner and purposes for simplifying rational expression calculations.Here is a typical sequence of steps of rational operations.

Example E.

a. Find the LCD of x + 1 x – 2 , x + 4

2x – 1

Clearing Denominator with LCDThe LCD is used for clearing denominators to simplify fractional calculations. Likewise the LCD is utilized in the same manner and purposes for simplifying rational expression calculations.Here is a typical sequence of steps of rational operations.

Example E.

Their LCD is (x – 2)(x + 4).

a. Find the LCD of x + 1 x – 2 , x + 4

2x – 1

Clearing Denominator with LCDThe LCD is used for clearing denominators to simplify fractional calculations. Likewise the LCD is utilized in the same manner and purposes for simplifying rational expression calculations.Here is a typical sequence of steps of rational operations.

Example E.

Their LCD is (x – 2)(x + 4).

a. Find the LCD of x + 1 x – 2 , x + 4

2x – 1

b. Simplifyx + 1 x – 2

(x – 2)(x + 4) ( )

Clearing Denominator with LCDThe LCD is used for clearing denominators to simplify fractional calculations. Likewise the LCD is utilized in the same manner and purposes for simplifying rational expression calculations.Here is a typical sequence of steps of rational operations.

Example E.

Their LCD is (x – 2)(x + 4).

x– 2 ( ) (x – 2)(x + 4)

a. Find the LCD of x + 1 x – 2 , x + 4

2x – 1

b. Simplifyx + 1 x – 2

(x – 2)(x + 4)

x + 1

( )

Clearing Denominator with LCDThe LCD is used for clearing denominators to simplify fractional calculations. Likewise the LCD is utilized in the same manner and purposes for simplifying rational expression calculations.Here is a typical sequence of steps of rational operations.

Example E.

Their LCD is (x – 2)(x + 4).

x– 2 ( ) (x – 2)(x + 4)

a. Find the LCD of x + 1 x – 2 , x + 4

2x – 1

b. Simplifyx + 1 x – 2

(x – 2)(x + 4)

x + 1

( )

Clearing Denominator with LCDThe LCD is used for clearing denominators to simplify fractional calculations. Likewise the LCD is utilized in the same manner and purposes for simplifying rational expression calculations.Here is a typical sequence of steps of rational operations.

Example E.

Their LCD is (x – 2)(x + 4).

x– 2 ( ) (x – 2)(x + 4)

a. Find the LCD of x + 1 x – 2 , x + 4

2x – 1

b. Simplifyx + 1 x – 2

(x – 2)(x + 4)

x + 1

( )

= (x + 1)(x + 4)

Clearing Denominator with LCDThe LCD is used for clearing denominators to simplify fractional calculations. Likewise the LCD is utilized in the same manner and purposes for simplifying rational expression calculations.Here is a typical sequence of steps of rational operations.

Example E.

Their LCD is (x – 2)(x + 4).

x– 2 ( ) (x – 2)(x + 4)

a. Find the LCD of x + 1 x – 2 , x + 4

2x – 1

b. Simplifyx + 1 x – 2

(x – 2)(x + 4)

x + 1

( )

= (x + 1)(x + 4)

= x2 + 5x + 4

Clearing Denominator with LCDThe LCD is used for clearing denominators to simplify fractional calculations. Likewise the LCD is utilized in the same manner and purposes for simplifying rational expression calculations.Here is a typical sequence of steps of rational operations.

Example E.

Their LCD is (x – 2)(x + 4).

x– 2 ( ) (x – 2)(x + 4)

a. Find the LCD of x + 1 x – 2 , x + 4

2x – 1

b. Simplifyx + 1 x – 2

x + 4 2x – 1

(x – 2)(x + 4)

c. Simplify (x – 2)(x + 4)

x + 1

( )

( )

= (x + 1)(x + 4)

= x2 + 5x + 4

Clearing Denominator with LCDThe LCD is used for clearing denominators to simplify fractional calculations. Likewise the LCD is utilized in the same manner and purposes for simplifying rational expression calculations.Here is a typical sequence of steps of rational operations.

Example E.

Their LCD is (x – 2)(x + 4).

x– 2 ( ) (x – 2)(x + 4)

a. Find the LCD of x + 1 x – 2 , x + 4

2x – 1

b. Simplifyx + 1 x – 2

x + 4 2x – 1

(x – 2)(x + 4)

c. Simplify (x – 2)(x + 4)

x + 1

( )

( )

= (x + 1)(x + 4)

= x2 + 5x + 4

x + 4 2x – 1

(x – 2)(x + 4) ( )

Clearing Denominator with LCDThe LCD is used for clearing denominators to simplify fractional calculations. Likewise the LCD is utilized in the same manner and purposes for simplifying rational expression calculations.Here is a typical sequence of steps of rational operations.

Example E.

Their LCD is (x – 2)(x + 4).

x– 2 ( ) (x – 2)(x + 4)

a. Find the LCD of x + 1 x – 2 , x + 4

2x – 1

b. Simplifyx + 1 x – 2

x + 4 2x – 1

(x – 2)(x + 4)

c. Simplify (x – 2)(x + 4)

x + 1

( )

( )

= (x + 1)(x + 4)

= x2 + 5x + 4

= (2x – 1)(x – 2)x + 4 2x – 1

(x – 2)(x + 4) ( )

Clearing Denominator with LCDThe LCD is used for clearing denominators to simplify fractional calculations. Likewise the LCD is utilized in the same manner and purposes for simplifying rational expression calculations.Here is a typical sequence of steps of rational operations.

Example E.

Their LCD is (x – 2)(x + 4).

x– 2 ( ) (x – 2)(x + 4)

a. Find the LCD of x + 1 x – 2 , x + 4

2x – 1

b. Simplifyx + 1 x – 2

x + 4 2x – 1

(x – 2)(x + 4)

c. Simplify (x – 2)(x + 4)

x + 1

( )

( )

= (x + 1)(x + 4)

= x2 + 5x + 4

= (2x – 1)(x – 2)

= 2x2 – 5x + 2 x + 4 2x – 1

(x – 2)(x + 4) ( )

Clearing Denominator with LCDThe LCD is used for clearing denominators to simplify fractional calculations. Likewise the LCD is utilized in the same manner and purposes for simplifying rational expression calculations.Here is a typical sequence of steps of rational operations.

x– 2 –

x + 4 ( )(x – 2)(x + 4)d. Simplify

x + 1 2x – 1

Clearing Denominator with LCD

x– 2 –

x + 4 )(x – 2)(x + 4)d. Simplify

x + 1 2x – 1

x – 2 –

x + 4 ( ) (x – 2)(x + 4)x + 1 2x – 1

Clearing Denominator with LCD

x– 2 –

x + 4 ( )(x – 2)(x + 4)d. Simplify

x + 1 2x – 1

(x + 4)

x – 2 –

x + 4 ( ) (x – 2)(x + 4)x + 1 2x – 1

Clearing Denominator with LCD

x– 2 –

x + 4 ( )(x – 2)(x + 4)d. Simplify

x + 1 2x – 1

(x + 4) (x – 2)

x – 2 –

x + 4 ( ) (x – 2)(x + 4)x + 1 2x – 1

Clearing Denominator with LCD

= (x + 1)(x + 4) – (2x – 1)(x – 2)

x– 2 –

x + 4 ( )(x – 2)(x + 4)d. Simplify

x + 1 2x – 1

(x + 4) (x – 2)

x – 2 –

x + 4 ( ) (x – 2)(x + 4)x + 1 2x – 1

Clearing Denominator with LCD

= (x + 1)(x + 4) – (2x – 1)(x – 2)

x– 2 –

x + 4 ( )(x – 2)(x + 4)d. Simplify

x + 1 2x – 1

(x + 4) (x – 2)

x – 2 –

x + 4 ( ) (x – 2)(x + 4)x + 1 2x – 1

= x2 + 5x + 4 – [2x2 – 5x + 2]

Clearing Denominator with LCD

= (x + 1)(x + 4) – (2x – 1)(x – 2)

x– 2 –

x + 4 ( )(x – 2)(x + 4)

Clearing Denominator with LCD

d. Simplify x + 1 2x – 1

(x + 4) (x – 2)

x – 2 –

x + 4 ( ) (x – 2)(x + 4)x + 1 2x – 1

= x2 + 5x + 4 – [2x2 – 5x + 2]

= x2 + 5x + 4 – 2x2 + 5x – 2

= (x + 1)(x + 4) – (2x – 1)(x – 2)

x– 2 –

x + 4 ( )(x – 2)(x + 4)

Clearing Denominator with LCD

d. Simplify x + 1 2x – 1

(x + 4) (x – 2)

x – 2 –

x + 4 ( ) (x – 2)(x + 4)x + 1 2x – 1

= x2 + 5x + 4 – [2x2 – 5x + 2]

= x2 + 5x + 4 – 2x2 + 5x – 2

= – x2 + 10x + 2

= (x + 1)(x + 4) – (2x – 1)(x – 2)

x– 2 –

x + 4 ( )(x – 2)(x + 4)

Clearing Denominator with LCD

d. Simplify x + 1 2x – 1

(x + 4) (x – 2)

x – 2 –

x + 4 ( ) (x – 2)(x + 4)x + 1 2x – 1

= x2 + 5x + 4 – [2x2 – 5x + 2]

= x2 + 5x + 4 – 2x2 + 5x – 2

= – x2 + 10x + 2

x– 2 –

x + 4 x + 1 2x – 1

e. Combine

= (x + 1)(x + 4) – (2x – 1)(x – 2)

x– 2 –

x + 4 ( )(x – 2)(x + 4)

Clearing Denominator with LCD

d. Simplify x + 1 2x – 1

(x + 4) (x – 2)

x – 2 –

x + 4 ( ) (x – 2)(x + 4)x + 1 2x – 1

= x2 + 5x + 4 – [2x2 – 5x + 2]

= x2 + 5x + 4 – 2x2 + 5x – 2

= – x2 + 10x + 2

x– 2 –

x + 4 x + 1 2x – 1

This is the next topic.

e. Combine

Ex. A. Find the LCM of the given expressions.

Methods of Finding LCM

xy3, xy1. x2y, xy32. xy3, xyz3.

4xy3, 6xy4. x2y, xy3, xy25. 9xy3, 12x3y, 15x2y36.

(x + 2), (x – 1) 7. (x + 2), (2x – 4) 8. (4x + 12), (10 – 5x) 9.

(4x + 6), (2x + 3) 10. (9x + 18), (2x + 4) 11. (–x + 2), (4x – 8) 12.

x2(x + 2), (x + 2)13. (x – 1)(x + 2), (x + 2)(x – 2)14.

(x – 3)(x – 3), (3 – x)(x + 2)17. x2(x – 2), (x – 2)x18.

19. x2(x + 2), (x + 2)(–x – 2) 20. (x – 1)(1 – x), (–1 – x)(x + 1)

(3x + 6)(x – 2), (x + 2)(x + 1)15. (x – 3)(2x + 1), (2x + 1)(x – 2)16.

x2 – 4x + 4, x2 – 421. x2 – x – 6, x2 – x – 222.

x2 + 2x – 3, 1 – x225. –x2 – 2x + 3, x2 – x26.

x2 – x – 6, x2 – 923. x2 + 5x – 6, x2 – x – 624.

2x2 + 5x – 3, x – 4x327. x2 – 5x + 4, x3 – 16x28.

Multiplication and Division of Rational Expressions

33. x – 2 x2 – 9

( –x + 1

x2 – 2x – 3 )

34. x + 3 x2 – 4

( –2x + 1

x2 + x – 2 )

35. x – 1 x2 – x – 6

( –x + 1

x2 – 2x – 3 )

36.x + 2

x2 – 4x + 3( – 2x + 1

x2 + 2x – 3 )

29. 4 – xx – 3

( –x – 1

2x + 3 ) * LCD

30. 3 – x x + 2

( –2x + 3 x – 3

)

Ex B. Find the LCD of the terms then expand and simplify

the following expressions. (These are from the last section.)

31. 3 – 4xx + 1

( –1 – 2x x + 3

)

32. 5x – 7 x + 5

( –4 – 5xx – 3

)

* LCD

* LCD

* LCD

* LCD

* LCD

* LCD

* LCD