Factoring Numbers

The ability to factor a number is an important skill to learn. You will be required to come up with all the factors of a number quickly when doing more complicated algebra later on in school. This lesson will get you up to speed on the basic ideas of factoring.

A factor of a number is one that divides into the number evenly. That is, 6 is a factor of 12 because 12 divided by 6 is exactly 2. The number 5 is not a factor of 12, because 12 divided by 5 is 2.4.

It is easy to find all the factors of a small number, like 3. The only numbers that divide evenly into 3 are 1 and 3. Finding the factors of an enormous number, like 64,448, can be very hard, because there could be several hundred factors. You will likely need to find factors of numbers in between, like 42. The first step is to recognize that 42 is an even number, and is divisible by 2. That operation reveals another factor, 21.

Since 2 goes evenly into 42, the result must also be true. 21 goes into 42 twice. Now we have four factors: 42, 21, 2, and 1, since the number and 1 are always factors of any number. We know that there aren't any more numbers that divide into the 2, but 21 is 7 x 3. That means that 7 and 3 are also factors.

After dividing 42 by 7 and 3, we discovered the last two factors, 6 and 14. We now have a total of 8 factors: 42, 21, 14, 7, 6, 3, 2, 1. No combinations of those numbers will give us any more factors, so we must be finished factoring 42.

That number was easy enough to factor because it was even, so we could start with 2. What about a more difficult number, like 81? The best idea is to try a few small odd numbers, like 3,5 and 7, to see if any of them divide evenly. It is also a good idea to check the number to see if it is prime, in which case it would only have 2 factors: itself and 1.

Sure enough, 81 is divisible by 3, giving 27 as an answer. That means that atleast 1, 3, 27, and 81 are factors of 81. We know that 27 is 9 x 3, so 9 is an additional factor. No other numbers work, so 1, 3, 9, 27, and 81 are the only 5 factors of 81.

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Imaginary Numbers

What is the square root of a negative number?

 Did you know that no real number multiplied by itself will ever produce a negative number?

Finding the square root of 4 is simple enough: either 2 or -2 multiplied by itself gives 4. However, there is no simple answer for the square root of -4.

So, what do you do when a discriminant is negative and you have to take its square root? This is where imaginary numbers come into play. Essentially, mathematicians have decided that the square root of -1 should be represented by the letter i. So, i = sqrt(-1), or you can write it this way: -1 1/2 or you can simply say: i 2 = -1.

What you should know about the number i:

1) i is not a variable.
2) i is not found on the real number line.
3) i is not a real number.

Sample A:

Simplify (4i) 2

Steps:

1) Multiply 4i times 4i. This will produce 16(i 2 ).

2) Multiply 16 times -1 because i2 equals -1.

The answer is: -16.

Sample B:

Simplify sqrt(-80).

Steps:

1) Multiply two radicands keeping in mind that one of them has to be a perfect square. How about sqrt(16) times sqrt(5)? Yes, this will produce sqrt(80). Also, don't forget to multiply sqrt(-1) times sqrt(16) times sqrt(5).

2) Simplify square roots where needed. For example, sqrt(16) becomes simply 4 and sqrt-1 simply becomes the number i.

3) Put it all together this way: 4i(sqrt(5)) or 4i times the square root of 5.

NOTE: You cannot reduce sqrt5 anymore because it is already in lowest terms.




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Greatest Common Factor

The greatest common factor is, by definition, the largest number that factors evenly into two or more larger numbers. For instance, the greatest common factor (GCF) of 15 and 25 is 5, because 5 is the largest number that goes into 15 and 25 evenly.

To find the GCF of small numbers, like 12 or 16, it might be easier to just list all the factors and find the largest common factor, but for big numbers like 490 and 819, you need a faster method. The first step is to complete a prime factorization of each number. Find any number (2 will work) that divides 490 evenly, and you will get 245. Keep dividing and keeping track of the numbers in a format like this:

Therefore, 7 * 7 * 5 * 2 = 490. Those four numbers circled in red are all prime and cannot be factored anymore, so you must be done. That is the prime factorization of 490. Now you can do the same with your other number, 819.

The prime factors of 819 are 3,3,7, and 13. Now we should compare the prime factors of both numbers:

819 = 3 * 3 * 7 * 13
490 = 2 * 5 * 7 * 7

What we want to do is take everything that is shared by both numbers. Only one factor is common, and that is 7, so we know that 7 is the GCF of 490 and 819. Let's try another pair of numbers, 1012 and 10580. Prime factorization of the numbers reveals this situation:

1012 = 2 * 2 * 11 * 23
10580 = 2 * 2 * 5 * 23 * 23

Each number has two 2's and one 23 in common, so we will use those numbers. We cannot use the other 23 because 1012 only has one, and we cannot use 11 because 10580 doesn't have any. Write down the numbers in common and multiply them to get the GCF:

2 * 2 * 23 = 92

That's all there is to finding the Greatest Common Factor (GCF) of a number. If you need more help, please visit another lesson on greatest common factor browse our site, or use a search engine like Google to find more information.

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Exponential Functions

What is an exponential function?

An exponential function is a mathematical expression in which a variable represents the exponent of an expression.

What does an exponential function look like?

Here's a very simple exponential function:

That equation is read as "y equals 2 to the x power."

Exponent refresher:

Let's remember how exponents work. Suppose we have the equation below:

That equation tells us to multiply x by itself to get y. It's the equivalent of:

If we want to find y when x=3, we can pretty quickly find that y=3*3=9. But, this is actually what's known as a "power function". In fact, it's just a polynomial, and not an exponential function at all.

Take a closer look at x2 . This means x squared or x to the second power. What does it mean?

We have two parts here:

1) An exponent,which is the number 2.
2) A base, which is the variable x.

With exponential functions, the variable will actually be the exponent, with a constant as the base.

Exponential Functions

Here's what exponential functions look like:

The equation is y equals 2 raised to the x power. This sort of equation represents what we call "exponential growth" or "exponential decay." Other examples of exponential functions include:

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Division of Rational Functions

Just like rational number division (division of regular fractions), multiply the inverse or the reciprocal. This process is also called "INVERT AND MULTIPLY." For example, suppose you had to divide 1/2 by 3/7. The typical procedure reminds us to "never mind the reason why, just invert and multiply." So following that rule you multiply 1/2 times 7/3 to arrive at the answer of 7/6. This same procedure can be used to divide rational functions. 

*Sample

(x + 1)/(x + 3) DIVIDED by (3x + 3)/(x - 2)

1) Invert right side fraction.

The right side fraction should then look like this: (x - 2)/(3x + 3).

2) Replace division symbol with multiplication symbol (because never mind the reason why, just invert and multiply).

3) Multiply numerator by numerator and denominator by denominator using the FOIL Method.

Numerator: (x + 1) ( x - 2) becomes x2 - x - 2

Denominator (x + 3) (3x + 3) becomes 3x2 + 12x + 9

4) Reduce fraction (if needed)

Final answer: (x 2 - x - 2)/(3x2 + 12x + 9)

Distributive Property

The distributive property is actually a very simply concept to learn and apply. It will allow you to simplify something like 3(6x + 4), where you have a number being multiplied by a set of parenthesis. Let's start with a simple problem:

6(4 + 2)

Based on the order of operation you know that anything inside parenthesis should be done first. Adding 4 + 2 is simple enough, resulting in this:

6(6)

When you see a number next to parenthesis like this, it means multiplication, so what we really have here is this (remember that * means multiplication):

6 * 6 = 36

That was easy enough, but what about a more difficult problem? Let's suppose that the 4 was really 4x, meaning 4 times the variable x. The distributive property allows you to simplify an expression like this, where you cannot just do the parenthesis and multiply.

6(4x + 2)

What this expression seems to say is that we want 6 times the sum of 4x + 2. It can also be expressed in a different way using the distributive property:

(6 * 4x) + (6 * 2)

We can do this because with 6(4x + 2), the 6 is distributed to the 4x and the 2. That expression can now be simplified to 24x + 12, which is easier to use that the original 6(4x + 2). Now try simplifying this expression:

-2(4y - 8)

This is no more difficult so simplify than the last one. Just distribute the -2 to the 4y and the -8:

(-2 * 4y) + (-2 * -8)

-8y + 16

16 - 8y

And that's all there is to it. Once you get the hang of things it will be second-nature to you. You are welcome to continue browsing our site now, or you can read another lesson on the distributive property from AlgebraHelp or this lesson from Dr. Math

Direct Variation

When two variables are related in such a way that the ratio of their values always remains the same, the two variables are said to be in direct variation.

In simpler terms, that means if A is always twice as much as B, then they directly vary. If a gallon of milk costs $2, and I buy 1 gallon, the total cost is $2. If I buy 10 gallons, the price is $20. In this example the total cost of milk and the number of gallons purchased are subject to direct variation -- the ratio of the cost to the number of gallons is always 2.

To be more "mathematical" about it, if y varies directly as x, then the graph of all points that describe this relationship is a line going through the origin (0, 0) whose slope is called the constant of the variation. That's because each of the variables is a constant multiple of the other, like in the graph shown below:

RADICALS AND VARIATION

RADICALS.......
            There are 5 radical 


*Roots of numbers and rational exponents
*Simplifying radical expression
*Addition and subtraction of radicals
*Multiplication and division of radicals
*Radical equation
   
 VARIATION......
   There are only 3 variation


*Direct variation
*Inverse variation
*Joint and combined variation

APPLICATION OF SPECIAL PRODUCTS AND FACTORING

Connect .. 

  * A number of problem situations can be translated to polynomial equations, which can in turn be solved by special products and factoring . 

..Explore ..  

  - In a basketball league of x teams in which every team plays every other team twice, the total number of games played is x - x

FACTORING GENERAL TRINOMIALS

You will find that factoring general trinomials requires skill, experience, and often, trial and error.

In this lesson , we will use capital letters A,B, and C to stand for the coefficients of the troinomial instead of the usual a,b, and c.




* FACTORING TRINOMIAL'S  OF THE FORM x + Bx + C


   - Let us begin by verifying the concept of factoring trinomials geometrically using algebra tiles. Recall that the dimension of a rectangle represent the factors and the area of the rectangle represents the product .

*FACTORING TRINOMIALS OF THE FORM Ax + Bx + C


 - When the coefficient of the x term is not 1 , there are no possibilities to consider.  We will consider three methods of factoring this trinomial.

Math Pictures

Father of Math

Archimedes took plain geometry to a higher level by considering solid geometry such as cones and spheres.Archimedes is especially important for his discovery of the relation between the surface and volume of a sphere and its circumscribing cyclinder.  Archimedes also famous for his theorem on displacement of water from a solid body.  Other than solid geometry, Archimedes also worked on mechanics and hydraulics.  Methods used by Archimedes was used by Newton to develop calculus.  Mathematics of Archimedes is used for many diverse fields ranging from Engineering, Hydraulic, Mechanics and Chemical Engineering.  As per Greek historian Proclus of 4th century AD, Roman soldiers would run away when they see a rope or a lever fearing that "Archimedes has deviced another machine to kill them".  This great mathematician was killed by a Roman soldier while working on a problem on the sand.

Math History

The area of study known as the history of mathematics is primarily an investigation into the origin of discoveries in mathematics and, to a lesser extent, an investigation into the mathematical methods and notation of the past.
Before the modern age and the worldwide spread of knowledge, written examples of new mathematical developments have come to light only in a few locales. The most ancient mathematical texts available are Plimpton 322 (Babylonian mathematics c. 1900 BC), the Rhind Mathematical Papyrus (Egyptian mathematics c. 2000-1800 BC) and the Moscow Mathematical Papyrus (Egyptian mathematics c. 1890 BC). All of these texts concern the so-called Pythagorean theorem, which seems to be the most ancient and widespread mathematical development after basic arithmetic and geometry.
The study of mathematics as a subject in its own right begins in the 6th century BC with the Pythagoreans, who coined the term "mathematics" from the ancient Greek μάθημα (mathema), meaning "subject of instruction". Greek mathematics greatly refined the methods (especially through the introduction of deductive reasoning and mathematical rigor in proofs) and expanded the subject matter of mathematics. Chinese mathematics made early contributions, including a place value system. The Hindu-Arabic numeral system and the rules for the use of its operations, in use throughout the world today, likely evolved over the course of the first millennium AD in India and was transmitted to the west via Islamic mathematics. Islamic mathematics, in turn, developed and expanded the mathematics known to these civilizations. Many Greek and Arabic texts on mathematics were then translated into Latin, which led to further development of mathematics in medieval Europe.
From ancient times through the Middle Ages, bursts of mathematical creativity were often followed by centuries of stagnation. Beginning in Renaissance Italy in the 16th century, new mathematical developments, interacting with new scientific discoveries, were made at an increasing pace that continues through the present day.