Math Tutorial - Finding the Greatest Common Factor and Factoring Squares

Greatest Common Factor: of two numbers is the result of two numbers being factored into their smaller factors individually and the of all the numbers that are factors, the one that is greatest is the greatest common factor.

Symbol: ( ) means greatest common factor.

For instance, (6,8) means the greatest common factor of 6 and 8.

Example: (6, 8) is,

First, find all the factors of 6. The factors of 6 are all the numbers that when multiplied by another number give us 6. Those are, 1, 2, 3, 6. This is so because:

1*6=6 and 2*3=6

So you see that each number when multiplied by another number gives us 6.

Similarly,

Second, Find the factors of 8.

Those are, 1, 2, 4, 8.

So, now, when we look at the factors of 6 and 8

for 6: 1, 2, 3, 6

for 8: 1, 2, 4, 8

We see that 1, and 2, both appear in the factors of both six and 8:

Now, of the factors that appear in both numbers, that is 1 and 2, the greatest one is 2. *

We use greatest instead of greater because when we speak of bigger numbers, there might be more than just two numbers that are common factors.

Factoring A Difference Of Two Squares

Once students are familiar with the idea of factoring, there are shortcuts to the FOIL method of quadratic equation factoring. One of these is factoring a difference of two squares. A difference of two squares means that we have a monomial multiplied to another monomial to give us a quadratic function where one monomial is a conjugate pair of the other and the middle term in the quadratic form of an equation disappears.

A quadratic equation is represented by ax^2+bx+c, when the "bx" term disappears, we have a difference of two squares.

Example:

(x-3)*(x+3), here (x+3) is a conjugate pair of (x-3), and here is where we get a difference of two squares.

(x-3)(x+3), using the FOIL method, is:

(x)(x)+(x)(3)+(-3)(x)+(-3)(3)=

x^2+3x-3x-9

x^2-9.

Here, we see that since the middle term disappeared, we see that (x-3)(x+3) is simply X^2-(3)^2, the two squares being X and 3, which is why it is called the difference of two squares.

Of course, the same thing would result if we were doing (x+3)(x-3) since multiplication is commutative.

Now, to give more examples:

(x+4)(x-4)= (x^2-4^2)=(x^2-16)

(x+9)(x-9)=(x^2-9^2)=(x^2-81)

And so on, from here on, the patern is too clear and it would be too much repetition to go on.

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Algebra for Beginners - How To Factor and Use the Difference of Two Squares

Formulas are an important part of all math classes because they state relationships that are ALWAYS true, and they generally make various mathematical tasks easier to perform. Factoring is one of those fundamental tasks in Algebra. Factoring allows us to reduce algebraic fractions into simpler form, and it can help us solve equations. Factoring the difference of two squares is one of the most commonly used processes in all of Algebra. Understanding when and how to use it is critical to success in Algebra.

We have already learned the meaning of "to factor," but it is always a good idea to review the definition. Factoring is the process of re-writing an expression using multiplication.

Before we can factor the difference of two squares, we need to be able to identify it. What exactly is a difference of two squares? To fully understand, let's look at each word. "Difference" means subtraction, but subtraction of what? "Two" tells us that we have two numbers and/or algebraic expressions. Thus far, we know we are going to subtract one number or expression from another; but these numbers are special. Our two numbers or expressions are perfect squares, like 1, 4, 9, 16, 25, 36, 49, etc and/or a^2, b^4, x^2, (xy)^2, etc. A "difference of two squares" will look like 25 - 9 or x^2 - y^4. Now, we are ready for the actual formula.

In symbols: a^2 - b^2 = (a + b)(a - b)

In words: The difference of the squares of two numbers factors as the product of the sum and difference of those numbers.

Note: It is extremely important that you be able to state these definitions out loud and that you understand every word. Don't move on until you know you are ready.

Before we actually use this formula, let's make certain it is true. While this is not a formal proof, we are going to test this formula with a number example like 25 - 9. (Both 25 and 9 are perfect squares.) By our formula, since 25 = 5^2 and 9 = 3^2, 25 - 9 must be equal to (5 + 3)(5 - 3). So, is the formula true? 25 - 9 = 16 by just doing order of operations. (5 + 3)(5 - 3) = (8)(2) = 16. Both expressions have the value 16. Again, I caution that this is not a proof. Since the proof is not the point of this article, I will ask that you either trust me or do several more examples to convince yourself of the validity of this formula.

About now, you ought to be thinking, "Why would I want to do that?" It is easier to evaluate 25 - 9 than it is to evaluate (5 + 3)(5 - 3); but keep in mind that we will primarily be using this relationship for the purpose of reducing algebraic fractions and solving algebraic equations.

For example: Solve the equation x^2 = 16.

Many students will quickly jump to the "answer" of 4 since 4^2 is 16. However, this equation has two answers, but it is not obvious where the other answer comes from. Noticing that both x^2 and 16 are perfect squares, we should think about the possibility of a difference of two squares. We can rewrite x^2 = 16 as x^2 - 16 = 0. Now we have a difference of two squares that factors as (x + 4)(x - 4) = 0. The two different factors produce the two solutions, x = 4 and x = -4.

To make best use of this strategy, you must form a new habit. Every time you encounter an equation of the form x^2 = a number, take the time to re-write that equation. Thus, you must see a^2 = 121 as a^2 - 121 = 0. This, then, can be factored as (a + 11)(a - 11) = 0 for the two solutions of a = +11 and a = -11.

Forming a habit of constantly looking for a difference of two squares can make reducing algebraic fractions a much simpler process.

For example: If possible, reduce the fraction (x^2 - 25) / (x + 5).

We can recognize a difference of two squares in the numerator, so factoring it should be automatic. This produces (x + 5)(x - 5) / ( x + 5). Reducing the common factors of (x + 5) leaves the final reduced result of x - 5.

Why does it matter that x - 5 is the reduced version of (x^2 - 25) / (x + 5)? There are two different reasons for doing this. First, let's look at these two equations: (x^2 - 25) / (x + 5) = 12 and x - 5 = 12. These two equations are equivalent, but which one would you rather solve? Could you do the first one in your head? Probably not. But the second equation has the obvious answer of x = 17.

The second reason for using the difference of two squares is that it makes evaluation of expressions for a given value much simpler. We already know that (x^2 - 25) / (x + 5 ) is the same as x - 5. Now, let's pretend that x has the value of 13 and I ask you to evaluate (x^2 - 25) / (x + 5) for x = 13. The unsimplified version becomes (13^2 - 25) / (13 + 5) or (169 - 25) / (13 + 5) or 144/18 = 8. That was a lot of work. But evaluating x - 5 with x = 13 is simple: 13 - 5 = 8.

With only a little practice, you will be able to simplify and evaluate expressions like the one above in your head. The ability to recognize and factor the difference of two squares will make your life in Algebra class much easier.

Shirley Slick, "The Slick Tips Lady," is a retired high school math teacher and tutor with degrees in Mathematics and Psychology and additional training in brain-based learning/teaching. Her goals: (1) to help parents help their children with math, (2) to help eliminate the horrendous Algebra failure rate, and (3) to inform the general public about problematic issues related to the field of education. For your free copy of "10 Slick Tips for Improving Your Child's Study Habits," visit her website at http://myslicktips.com/

Algebra For Beginners - What Does It Means To Factor and Why Is It So Important In Algebra?

Algebra has become a "must take" course in high school. However, while technology keeps advancing and Algebra becomes a course in which everyone needs to be proficient, student success rates remain consistently low. There are many reasons for this, but one of those reasons is that students do not become proficient in the vocabulary part of mathematics. In fact, math students complain loudly if given vocabulary and spelling quizzes, claiming, "this isn't English class!" Students fail to understand that if they do not master the vocabulary, they will perform badly on tests because they won't understand what the directions are telling them to do. The instruction "Factor..." is a prime example of this problem.

What Is Factoring?

In simplest terms, the process of "factoring" or the verb "to factor" means to re-write an algebraic expression in terms of multiplication. Factoring is a process that is used throughout Algebra; but if you ask even the top students what the verb "to factor" means, very few will answer correctly.

Students understand factoring from arithmetic. Ask students to "factor 6," they will know to write 6 = 2 x 3. Asking students to "factor 12" may result in 12 = 2 x 6 or 12 = 3 x 4, and a few students might continue factoring to 12 = 2 x 6 = 2 x 2 x 3 = 2^2 x 3. These students know to factor to a product of prime numbers.

However, asking an Algebra class early in the year to factor the expression 2x + 2y will get you a room full of blank stares. They do not transfer their knowledge of the word "factor" from arithmetic to Algebra. For Algebra students to get proficient at the terminology, we need to give our students many examples of what we are doing; and we need to have our students saying and explaining definitions and properties and giving examples OUT LOUD.

Initially, factoring in Algebra relies heavily on the Distributive Property. Surprisingly, students seem to quickly understand and effectively use the distributive property for factoring two uncomplicated terms. When looking at 2x + 2y, students generally recognize the common multiplier of 2. Then they learn to use the Distributive Property to re-write the expression using multiplication. 2x + 2y = 2(x + y).

Students have a little more difficulty as the terms get more complicated and/or the number of terms increases; but with several examples and practice, students can factor expressions like: 3a + 9ab - 15ac. Each term shares both 3 and a. Again, using the Distributive Property, the expression 3a + 9ab - 15ac becomes 3a(1 + 3b - 5c) when re-written as multiplication.

Note: One of the best things about factoring is that it can be easily checked by performing the multiplication to verify that the result is the original expression.

Now that we know WHAT factoring is, we need to understand the reasons WHY to factor.

Why Do We Need To Factor?

There are two main uses for factoring: (1) reducing fractions, and (2) solving equations.

Reducing Fractions:

As with factoring, students learn to reduce fractions in arithmetic. 12/14 = (2 x 6)/(2 x 7) = (2/2)(6/7) = 1(6/7) = 6/7.

Making the transfer of that knowledge to Algebra often causes trouble; but just as in arithmetic, reducing algebraic fractions uses factoring and the fact that x/x = 1.

Reduce: (3a + 9ab) / (3a^2 + 15ab).

This fraction becomes 3a(1 + 3b) / 3a(a + 5b) = (3a/3a) ((1 + 3b)/(a + 5b)) = 1 ((1 + 3b)/(a + 5b)) = (1 + 3b)/(a + 5b).

Solving Equations:

Some algebraic equations can be solved (which means to find the values that make the equation true) by first moving all the terms to one side of the equal sign. This leaves zero on the other side. Then, if possible, the algebraic expression is factored. Then the fact that if ab = 0, then either a = 0 or b = 0 allows us to find solutions.

Solve: a^2 = 3a.

This equation becomes a^2 - 3a = 0. Then factoring gives us a(a - 3) = 0. So either a = 0 or (a - 3) = 0. This tells that we have two values that make the original equation true: 0 and 3.

For every example in this article, the factoring method used has been the Distributive Property. There are, however, other methods for factoring different types of expressions. Those will be discussed in other articles. For now, it is important that you remember:

(1) To factor means to re-write as multiplication, and

(2) Factoring is important because it is used to reduce algebraic fractions and to solve equations.

Shirley Slick, "The Slick Tips Lady," is a retired high school math teacher and tutor with degrees in Mathematics and Psychology and additional training in brain-based learning/teaching. Her goals: (1) to help parents help their children with math, (2) to help eliminate the horrendous Algebra failure rate, and (3) to inform the general public about problematic issues related to the field of education. For your free copy of "10 Slick Tips for Improving Your Child's Study Habits," visit her website at http://myslicktips.com/

Math Tutorial - Finding the Greatest Common Factor and Factoring Squares

Greatest Common Factor: of two numbers is the result of two numbers being factored into their smaller factors individually and the of all the numbers that are factors, the one that is greatest is the greatest common factor.

Symbol: ( ) means greatest common factor.

For instance, (6,8) means the greatest common factor of 6 and 8.

Example: (6, 8) is,

First, find all the factors of 6. The factors of 6 are all the numbers that when multiplied by another number give us 6. Those are, 1, 2, 3, 6. This is so because:

1*6=6 and 2*3=6

So you see that each number when multiplied by another number gives us 6.

Similarly,

Second, Find the factors of 8.

Those are, 1, 2, 4, 8.

So, now, when we look at the factors of 6 and 8

for 6: 1, 2, 3, 6

for 8: 1, 2, 4, 8

We see that 1, and 2, both appear in the factors of both six and 8:

Now, of the factors that appear in both numbers, that is 1 and 2, the greatest one is 2. *

We use greatest instead of greater because when we speak of bigger numbers, there might be more than just two numbers that are common factors.

Factoring A Difference Of Two Squares

Once students are familiar with the idea of factoring, there are shortcuts to the FOIL method of quadratic equation factoring. One of these is factoring a difference of two squares. A difference of two squares means that we have a monomial multiplied to another monomial to give us a quadratic function where one monomial is a conjugate pair of the other and the middle term in the quadratic form of an equation disappears.

A quadratic equation is represented by ax^2+bx+c, when the "bx" term disappears, we have a difference of two squares.

Example:

(x-3)*(x+3), here (x+3) is a conjugate pair of (x-3), and here is where we get a difference of two squares.

(x-3)(x+3), using the FOIL method, is:

(x)(x)+(x)(3)+(-3)(x)+(-3)(3)=

x^2+3x-3x-9

x^2-9.

Here, we see that since the middle term disappeared, we see that (x-3)(x+3) is simply X^2-(3)^2, the two squares being X and 3, which is why it is called the difference of two squares.

Of course, the same thing would result if we were doing (x+3)(x-3) since multiplication is commutative.

Now, to give more examples:

(x+4)(x-4)= (x^2-4^2)=(x^2-16)

(x+9)(x-9)=(x^2-9^2)=(x^2-81)

And so on, from here on, the patern is too clear and it would be too much repetition to go on.

Algebra for Beginners - How To Factor and Use the Difference of Two Squares

Formulas are an important part of all math classes because they state relationships that are ALWAYS true, and they generally make various mathematical tasks easier to perform. Factoring is one of those fundamental tasks in Algebra. Factoring allows us to reduce algebraic fractions into simpler form, and it can help us solve equations. Factoring the difference of two squares is one of the most commonly used processes in all of Algebra. Understanding when and how to use it is critical to success in Algebra.

We have already learned the meaning of "to factor," but it is always a good idea to review the definition. Factoring is the process of re-writing an expression using multiplication.

Before we can factor the difference of two squares, we need to be able to identify it. What exactly is a difference of two squares? To fully understand, let's look at each word. "Difference" means subtraction, but subtraction of what? "Two" tells us that we have two numbers and/or algebraic expressions. Thus far, we know we are going to subtract one number or expression from another; but these numbers are special. Our two numbers or expressions are perfect squares, like 1, 4, 9, 16, 25, 36, 49, etc and/or a^2, b^4, x^2, (xy)^2, etc. A "difference of two squares" will look like 25 - 9 or x^2 - y^4. Now, we are ready for the actual formula.

In symbols: a^2 - b^2 = (a + b)(a - b)

In words: The difference of the squares of two numbers factors as the product of the sum and difference of those numbers.

Note: It is extremely important that you be able to state these definitions out loud and that you understand every word. Don't move on until you know you are ready.

Before we actually use this formula, let's make certain it is true. While this is not a formal proof, we are going to test this formula with a number example like 25 - 9. (Both 25 and 9 are perfect squares.) By our formula, since 25 = 5^2 and 9 = 3^2, 25 - 9 must be equal to (5 + 3)(5 - 3). So, is the formula true? 25 - 9 = 16 by just doing order of operations. (5 + 3)(5 - 3) = (8)(2) = 16. Both expressions have the value 16. Again, I caution that this is not a proof. Since the proof is not the point of this article, I will ask that you either trust me or do several more examples to convince yourself of the validity of this formula.

About now, you ought to be thinking, "Why would I want to do that?" It is easier to evaluate 25 - 9 than it is to evaluate (5 + 3)(5 - 3); but keep in mind that we will primarily be using this relationship for the purpose of reducing algebraic fractions and solving algebraic equations.

For example: Solve the equation x^2 = 16.

Many students will quickly jump to the "answer" of 4 since 4^2 is 16. However, this equation has two answers, but it is not obvious where the other answer comes from. Noticing that both x^2 and 16 are perfect squares, we should think about the possibility of a difference of two squares. We can rewrite x^2 = 16 as x^2 - 16 = 0. Now we have a difference of two squares that factors as (x + 4)(x - 4) = 0. The two different factors produce the two solutions, x = 4 and x = -4.

To make best use of this strategy, you must form a new habit. Every time you encounter an equation of the form x^2 = a number, take the time to re-write that equation. Thus, you must see a^2 = 121 as a^2 - 121 = 0. This, then, can be factored as (a + 11)(a - 11) = 0 for the two solutions of a = +11 and a = -11.

Forming a habit of constantly looking for a difference of two squares can make reducing algebraic fractions a much simpler process.

For example: If possible, reduce the fraction (x^2 - 25) / (x + 5).

We can recognize a difference of two squares in the numerator, so factoring it should be automatic. This produces (x + 5)(x - 5) / ( x + 5). Reducing the common factors of (x + 5) leaves the final reduced result of x - 5.

Why does it matter that x - 5 is the reduced version of (x^2 - 25) / (x + 5)? There are two different reasons for doing this. First, let's look at these two equations: (x^2 - 25) / (x + 5) = 12 and x - 5 = 12. These two equations are equivalent, but which one would you rather solve? Could you do the first one in your head? Probably not. But the second equation has the obvious answer of x = 17.

The second reason for using the difference of two squares is that it makes evaluation of expressions for a given value much simpler. We already know that (x^2 - 25) / (x + 5 ) is the same as x - 5. Now, let's pretend that x has the value of 13 and I ask you to evaluate (x^2 - 25) / (x + 5) for x = 13. The unsimplified version becomes (13^2 - 25) / (13 + 5) or (169 - 25) / (13 + 5) or 144/18 = 8. That was a lot of work. But evaluating x - 5 with x = 13 is simple: 13 - 5 = 8.

With only a little practice, you will be able to simplify and evaluate expressions like the one above in your head. The ability to recognize and factor the difference of two squares will make your life in Algebra class much easier.

Algebra For Beginners - What Does It Means To Factor and Why Is It So Important In Algebra?

Algebra has become a "must take" course in high school. However, while technology keeps advancing and Algebra becomes a course in which everyone needs to be proficient, student success rates remain consistently low. There are many reasons for this, but one of those reasons is that students do not become proficient in the vocabulary part of mathematics. In fact, math students complain loudly if given vocabulary and spelling quizzes, claiming, "this isn't English class!" Students fail to understand that if they do not master the vocabulary, they will perform badly on tests because they won't understand what the directions are telling them to do. The instruction "Factor..." is a prime example of this problem.

What Is Factoring?

In simplest terms, the process of "factoring" or the verb "to factor" means to re-write an algebraic expression in terms of multiplication. Factoring is a process that is used throughout Algebra; but if you ask even the top students what the verb "to factor" means, very few will answer correctly.

Students understand factoring from arithmetic. Ask students to "factor 6," they will know to write 6 = 2 x 3. Asking students to "factor 12" may result in 12 = 2 x 6 or 12 = 3 x 4, and a few students might continue factoring to 12 = 2 x 6 = 2 x 2 x 3 = 2^2 x 3. These students know to factor to a product of prime numbers.

However, asking an Algebra class early in the year to factor the expression 2x + 2y will get you a room full of blank stares. They do not transfer their knowledge of the word "factor" from arithmetic to Algebra. For Algebra students to get proficient at the terminology, we need to give our students many examples of what we are doing; and we need to have our students saying and explaining definitions and properties and giving examples OUT LOUD.

Initially, factoring in Algebra relies heavily on the Distributive Property. Surprisingly, students seem to quickly understand and effectively use the distributive property for factoring two uncomplicated terms. When looking at 2x + 2y, students generally recognize the common multiplier of 2. Then they learn to use the Distributive Property to re-write the expression using multiplication. 2x + 2y = 2(x + y).

Students have a little more difficulty as the terms get more complicated and/or the number of terms increases; but with several examples and practice, students can factor expressions like: 3a + 9ab - 15ac. Each term shares both 3 and a. Again, using the Distributive Property, the expression 3a + 9ab - 15ac becomes 3a(1 + 3b - 5c) when re-written as multiplication.

Note: One of the best things about factoring is that it can be easily checked by performing the multiplication to verify that the result is the original expression.

Now that we know WHAT factoring is, we need to understand the reasons WHY to factor.

Why Do We Need To Factor?

There are two main uses for factoring: (1) reducing fractions, and (2) solving equations.

Reducing Fractions:

As with factoring, students learn to reduce fractions in arithmetic. 12/14 = (2 x 6)/(2 x 7) = (2/2)(6/7) = 1(6/7) = 6/7.

Making the transfer of that knowledge to Algebra often causes trouble; but just as in arithmetic, reducing algebraic fractions uses factoring and the fact that x/x = 1.

Reduce: (3a + 9ab) / (3a^2 + 15ab).

This fraction becomes 3a(1 + 3b) / 3a(a + 5b) = (3a/3a) ((1 + 3b)/(a + 5b)) = 1 ((1 + 3b)/(a + 5b)) = (1 + 3b)/(a + 5b).

Solving Equations:

Some algebraic equations can be solved (which means to find the values that make the equation true) by first moving all the terms to one side of the equal sign. This leaves zero on the other side. Then, if possible, the algebraic expression is factored. Then the fact that if ab = 0, then either a = 0 or b = 0 allows us to find solutions.

Solve: a^2 = 3a.

This equation becomes a^2 - 3a = 0. Then factoring gives us a(a - 3) = 0. So either a = 0 or (a - 3) = 0. This tells that we have two values that make the original equation true: 0 and 3.

For every example in this article, the factoring method used has been the Distributive Property. There are, however, other methods for factoring different types of expressions. Those will be discussed in other articles. For now, it is important that you remember:

(1) To factor means to re-write as multiplication, and

(2) Factoring is important because it is used to reduce algebraic fractions and to solve equations.