10 Factoring Methods in Mathematics

The factoring Is a method used in mathematics to simplify an expression that can contain numbers, variables or a combination of both.

To talk about factorization, the student must first immerse themselves in the world of mathematics and understand certain basic concepts.

Constants and variables are two fundamental concepts. A constant is a number, which can be either. The beginner usually has problems to solve with whole numbers that are easier to handle, but later this field is extended to any real and even complex amount.

For its part, we are often told that the variable is the"x", and takes any value. But this concept is a bit bare. To assimilate it better, imagine that we travel an infinite road in a given direction.

Each moment of time we move through it and it is the distance traveled since we began our walk that tells us our position. Our position is the variable.

Now, if you walked 300 meters down that road, but I walked 600, I can say that my position is 2 times yours, or I = 2 * TU. The variables of the equation are TÚ and IO, and the constant is 2. This constant value is the factor that multiplies the variable.

When we have more complicated equations, we use factorization, which is to extract the factors that are common to simplify the expression, to make it easier to solve or to be able to do algebraic operations with it.

Factorization in prime numbers

A prime number is an integer which is divisible only by itself and by unity. Number one is not considered prime number.

The prime numbers are 2, 3, 5, 7, 11... etc. There is not a formula for calculating a prime number to date, so to know whether a number is prime or not, you should try to factor and test.

Factoring a number in prime numbers is to find the numbers that, multiplied and summed, give us the given number. For example, if we have the number 132, we decompose it as follows:

In this way, we have factorized 132 as the multiplication of prime numbers.

Polynomials

Let's go back to the road

Now not only you and I are walking down the road. There are other people too. Each one represents a variable. And not only do we continue to walk along the road, but some turn aside and leave the road. We walk in the plane and not on the straight.

To complicate it a little more, some people not only double or multiply by a factor our speed, but they could be as fast as the square or the cube or the nth power of ours.

To the new expression we will call it polynomial since it expresses many variables at the same time. The degree of the polynomial is given by the largest exponent of its variable.

Ten cases of factorization

1- To factorize a polynomial, we again look for common factors (which are repeated) in the expression.

2- It is possible that the common factor is in turn a polynomial, for example:

3- Perfect square trinomial. This is called the expression resulting from squaring a binomial.

4- Difference of perfect squares. Occurs when the expression is the subtraction of two terms that have exact square root:

5- Perfect square trinomial by addition and subtraction. It occurs when the expression has three terms; A pair of them are perfect squares and the third is completed with sum to be double the product of the roots.

It would be desirable for it to be of the form

Then we add the missing terms and subtract them, so as not to alter the equation:

Regrouping we have:

Now we apply the sum of squares that says:

Where:

6- Trinomial of the form:

In this case the following procedure is performed:

Example: be the polynomial

The polynomial will be decomposed into two factors in the form of binomials such as:

The sign will depend on the following: In the first of the factors, the sign will have the same of the second of the terms of the trinomial, in this case (+2); In the second of the factors, will have the sign result of multiplying the signs of the second and third factors of the trinomial (+12) (+ 36)) = + 432.

If the signs prove to be equal in both cases, we will look for two numbers that add up to the second term and the product or multiplication equals the third of the terms of the trinomial:

K + m = b; K.m = c

On the other hand, if the signs are not equal, two numbers must be found such that the difference is equal to the second term and its multiplication results in the value of the third term.

K-m = b; K.m = c

In our case:

Then the factorization is:

7- Trinomial of the form

Unlike the previous case, the coefficient of the quadratic term is multiplied by a coefficient other than one. In this case, proceed as follows. Example:

The whole trinomial is multiplied by the coefficient a.

The trinomial will be decomposed into two binomial-shaped factors whose first term is the root of the quadratic term

The numbers s and p are such that their sum is equal to coefficient 8 and its multiplication to 12

8- Sum or difference of nth powers. This is the case of the expression:

And the formula is applied:

In the case of power difference, regardless of whether n is even or odd, it applies:

Examples:

9- Perfect tetranomial cube. Already with the previous case, the following formulas are deduced:

10- Binomial divisors:

When we assume that a polynomial is the result of a multiplication of several binomials with each other, this method is applied. First, the zeros of the polynomial are determined.

Zeros or roots are the values ​​that make zero the equation. Each factor is created with the negative of the root found, for example, if the polynomial P (x) becomes zero for x = 8, then one of the binomials that compose it will be (x-8). Example:

The divisors of the independent term 14 are ± 1, ± 2, ± 7 and ± 14, so that it is evaluated to find if the binomials:

They are divisors of the polynomial.

Evaluating for each root:

The expression is then factorized as follows:

The polynomial is evaluated for the values:

All these methods of simplification are useful when solving practical problems in various areas whose principles are based on mathematical expressions such as physics, chemistry, etc., so they are vital tools in each of these sciences and their specific disciplines .

References

1. Integer Factorization. Retrieved from: academickids.com
2. Vilson, J. (2014). Edutopia: How to Teach Kids About Polynomial Factoring.
3. Fundamental Theorem of Arithmetic. Retrieved from: mathisfun.com.
4. The 10 cases of factorization. Retrieved from: teffymarro.blogspot.com.
5. Factoring Polynomials. Retrieved from: jamesbrennan.org.
6. Factoring third-degree polynomials. Retrieved from: blog.aloprofe.com.
7. How to factorize a cubic polynomial. Retrieved from: wikihow.com.
8. The 10 cases of factorization. Recovered from: taringa.net.