In mathematics, factorization (also factorisation in British English) or factoring is the decomposition of an object (for example, a number, a polynomial, or a matrix) into a product of other objects, or factors, which when multiplied together give the original. For example, the number 15 factors into primes as 3 × 5, and the polynomial x2 − 4 factors as (x − 2)(x + 2). In all cases, a product of simpler objects is obtained.
The aim of factoring is usually to reduce something to "basic building blocks," such as numbers to prime numbers, or polynomials to irreducible polynomials. Factoring integers is covered by the fundamental theorem of arithmetic and factoring polynomials by the fundamental theorem of algebra. Viète's formulas relate the coefficients of a polynomial to its roots.
The opposite of factorization is expansion. This is the process of multiplying together factors to recreate the original, "expanded" polynomial.
Integer factorization for large integers appears to be a difficult problem. There is no known method to carry it out quickly. Its complexity is the basis of the assumed security of some public key cryptography algorithms, such as RSA.
A matrix can also be factorized into a product of matrices of special types, for an application in which that form is convenient. One major example of this uses an orthogonal or unitary matrix, and a triangular matrix. There are different types: QR decomposition, LQ, QL, RQ, RZ.
Another example is the factorization of a function as the composition of other functions having certain properties; for example, every function can be viewed as the composition of a surjective function with an injective function. This situation is generalized by factorization systems.
Integers
By the fundamental theorem of arithmetic, every positive integer has a unique prime factorization. Given an algorithm for integer factorization, one can factor any integer down to its constituent primes by repeated application of this algorithm. For very large numbers, no efficient algorithm is known.
Quadratic polynomials
Any quadratic polynomial over the complex numbers (polynomials of the form ax2 + bx + c where a, b, and c ∈
can be factored into an expression with the form
using the quadratic formula. The method is as follows:
where α and β are the two roots of the polynomial, found with the quadratic formula.
Polynomials factorable over the integers
where
and
You can then set each binomial equal to zero, and solve for x to reveal the two roots. Factoring does not involve any other formulas, and is mostly just something you see when you come upon a quadratic equation.
Take for example 2x2 − 5x + 2 = 0. Because a = 2 and mn = a, mn = 2, which means that of m and n, one is 1 and the other is 2. Now we have (2x + p)(x + q) = 0. Because c = 2 and pq = c, pq = 2, which means that of p and q, one is 1 and the other is 2 or one is −1 and the other is −2. A guess and check of substituting the 1 and 2, and −1 and −2, into p and q (while applying pn + mq = b) tells us that 2x2 − 5x + 2 = 0 factors into (2x − 1)(x − 2) = 0, giving us the roots x = {0.5, 2}
Note: A quick way to check whether the second term in the binomial should be positive or negative (in the example, 1 and 2 and −1 and −2) is to check the second operation in the trinomial (+ or −). If it is +, then check the first operation: if it is +, the terms will be positive, while if it is −, the terms will be negative. If the second operation is −, there will be one positive and one negative term; guess and check is the only way to determine which one is positive and which is negative.
If a polynomial with integer coefficients has a discriminant that is a perfect square, that polynomial is factorable over the integers.
For example, look at the polynomial 2x2 + 2x − 12. If you substitute the values of the expression into the quadratic formula, the discriminant b2 − 4ac becomes 22 − 4 × 2 × −12, which equals 100. 100 is a perfect square, so the polynomial 2x2 + 2x − 12 is factorable over the integers; its factors are 2, (x − 2), and (x + 3).
Now look at the polynomial x2 + 93x − 2. Its discriminant, 932 − 4 × 1 × (−2), is equal to 8657, which is not a perfect square. So x2 + 93x − 2 cannot be factored over the integers.
Perfect Square Trinomials
Some quadratics can be factored into two identical binomials. These quadratics are called perfect square trinomials. Perfect square trinomials can be factored as follows:
and 
Sum/Difference of Two Squares
Another common type of algebraic factoring is called the difference of two squares. It is the application of the formula
to any two terms, whether or not they are perfect squares. If the two terms are subtracted, simply apply the formula. If they are added, the two binomials obtained from the factoring will each have an imaginary term. This formula can be represented as
For example, 4x2 + 49 can be factored into (2x + 7i)(2x − 7i).
A visual illustration of the identity (a + b)2 = a2 + 2ab + b2
Factoring by grouping
Another way to factor some polynomials is factoring by grouping. This is done by placing the terms in the polynomial into two or more groups, where each group can be factored by a known method. The results of these factorizations can sometimes be combined to make an even more simplified expression. For example, to factor the polynomial
Group similar terms,
Factor out Greatest Common Factor,
Factor out binomial
AC Method
If a quadratic polynomial has rational solutions, we can find p and q so that pq = ac and p + q = b. (If the discriminant is a square number these exist, otherwise we have irrational or complex solutions, and the assumption of rational solutions is not valid.)
The terms on top will have common factors that can be factored out and used to cancel the denominator, if it is not 1. As an example consider the quadratic polynomial:
Inspection of the factors of ac = 36 leads to 4 + 9 = 13 = b.
Factoring Other Polynomials
Sum/difference of two cubes
Another formula for factoring is the sum or difference of two cubes. The sum can be represented by
and the difference by
For example, x3 − 103 (or x3 − 1000) can be factored into (x − 10)(x2 + 10x + 100).
Factoring Numbers
The aim of factoring is usually to reduce something to "basic building blocks," such as numbers to prime numbers, or polynomials to irreducible polynomials. Factoring integers is covered by the fundamental theorem of arithmetic and factoring polynomials by the fundamental theorem of algebra. Viète's formulas relate the coefficients of a polynomial to its roots.
The opposite of factorization is expansion. This is the process of multiplying together factors to recreate the original, "expanded" polynomial.
Integer factorization for large integers appears to be a difficult problem. There is no known method to carry it out quickly. Its complexity is the basis of the assumed security of some public key cryptography algorithms, such as RSA.
A matrix can also be factorized into a product of matrices of special types, for an application in which that form is convenient. One major example of this uses an orthogonal or unitary matrix, and a triangular matrix. There are different types: QR decomposition, LQ, QL, RQ, RZ.
Another example is the factorization of a function as the composition of other functions having certain properties; for example, every function can be viewed as the composition of a surjective function with an injective function. This situation is generalized by factorization systems.
Integers
By the fundamental theorem of arithmetic, every positive integer has a unique prime factorization. Given an algorithm for integer factorization, one can factor any integer down to its constituent primes by repeated application of this algorithm. For very large numbers, no efficient algorithm is known.
Quadratic polynomials
Any quadratic polynomial over the complex numbers (polynomials of the form ax2 + bx + c where a, b, and c ∈
can be factored into an expression with the form
using the quadratic formula. The method is as follows:
where α and β are the two roots of the polynomial, found with the quadratic formula.
Polynomials factorable over the integers
where
and
You can then set each binomial equal to zero, and solve for x to reveal the two roots. Factoring does not involve any other formulas, and is mostly just something you see when you come upon a quadratic equation.
Take for example 2x2 − 5x + 2 = 0. Because a = 2 and mn = a, mn = 2, which means that of m and n, one is 1 and the other is 2. Now we have (2x + p)(x + q) = 0. Because c = 2 and pq = c, pq = 2, which means that of p and q, one is 1 and the other is 2 or one is −1 and the other is −2. A guess and check of substituting the 1 and 2, and −1 and −2, into p and q (while applying pn + mq = b) tells us that 2x2 − 5x + 2 = 0 factors into (2x − 1)(x − 2) = 0, giving us the roots x = {0.5, 2}
Note: A quick way to check whether the second term in the binomial should be positive or negative (in the example, 1 and 2 and −1 and −2) is to check the second operation in the trinomial (+ or −). If it is +, then check the first operation: if it is +, the terms will be positive, while if it is −, the terms will be negative. If the second operation is −, there will be one positive and one negative term; guess and check is the only way to determine which one is positive and which is negative.
If a polynomial with integer coefficients has a discriminant that is a perfect square, that polynomial is factorable over the integers.
For example, look at the polynomial 2x2 + 2x − 12. If you substitute the values of the expression into the quadratic formula, the discriminant b2 − 4ac becomes 22 − 4 × 2 × −12, which equals 100. 100 is a perfect square, so the polynomial 2x2 + 2x − 12 is factorable over the integers; its factors are 2, (x − 2), and (x + 3).
Now look at the polynomial x2 + 93x − 2. Its discriminant, 932 − 4 × 1 × (−2), is equal to 8657, which is not a perfect square. So x2 + 93x − 2 cannot be factored over the integers.
Perfect Square Trinomials
Some quadratics can be factored into two identical binomials. These quadratics are called perfect square trinomials. Perfect square trinomials can be factored as follows:
and 
Sum/Difference of Two Squares
Another common type of algebraic factoring is called the difference of two squares. It is the application of the formula
to any two terms, whether or not they are perfect squares. If the two terms are subtracted, simply apply the formula. If they are added, the two binomials obtained from the factoring will each have an imaginary term. This formula can be represented as
For example, 4x2 + 49 can be factored into (2x + 7i)(2x − 7i).
A visual illustration of the identity (a + b)2 = a2 + 2ab + b2
Factoring by grouping
Another way to factor some polynomials is factoring by grouping. This is done by placing the terms in the polynomial into two or more groups, where each group can be factored by a known method. The results of these factorizations can sometimes be combined to make an even more simplified expression. For example, to factor the polynomial
Group similar terms,
Factor out Greatest Common Factor,
Factor out binomial
AC Method
If a quadratic polynomial has rational solutions, we can find p and q so that pq = ac and p + q = b. (If the discriminant is a square number these exist, otherwise we have irrational or complex solutions, and the assumption of rational solutions is not valid.)
The terms on top will have common factors that can be factored out and used to cancel the denominator, if it is not 1. As an example consider the quadratic polynomial:
Inspection of the factors of ac = 36 leads to 4 + 9 = 13 = b.
Factoring Other Polynomials
Sum/difference of two cubes
Another formula for factoring is the sum or difference of two cubes. The sum can be represented by
and the difference by
For example, x3 − 103 (or x3 − 1000) can be factored into (x − 10)(x2 + 10x + 100).
Factoring Numbers
"Factors" are the numbers you multiply to get another number. For instance, the factors of 15 are 3 and 5, because 3×5 = 15. Some numbers have more than one factorization (more than one way of being factored). For instance, 12 can be factored as 1×12, 2×6, or 3×4. A number that can only be factored as 1 times itself is called "prime". The first few primes are 2, 3, 5, 7, 11, and 13. The number 1 is not regarded as a prime, and is usually not included in factorizations, because 1 goes into everything. (The number 1 is a bit boring in this context, so it gets ignored.)
You most often want to find the "prime factorization" of a number: the list of all the prime-number factors of a given number. The prime factorization does not include 1, but does include every copy of every prime factor. For instance, the prime factorization of 8 is 2×2×2, not just "2". Yes, 2 is the only factor, but you need three copies of it to multiply back to 8, so the prime factorization includes all three copies.
On the other hand, the prime factorization includes ONLY the prime factors, not any products of those factors. For instance, even though 2×2 = 4, and even though 4 is a divisor of 8, 4 is NOT in the PRIME factorization of 8. That is because 8 does NOT equal 2×2×2×4! This accidental over-duplication of factors is another reason why the prime factorization is often best: it avoids counting any factor too many times. Suppose that you need to find the prime factorization of 24. Sometimes a student will just list all the divisors of 24: 1, 2, 3, 4, 6, 8, 12, and 24. Then the student will do something like make the product of all these divisors: 1×2×3×4×6×8×12×24. But this equals 331776, not 24. So it's best to stick to the prime factorization, even if the problem doesn't require it, in order to avoid either omitting a factor or else over-duplicating one.
In the case of 24, you can find the prime factorization by taking 24 and dividing it by the smallest prime number that goes into 24: 24 ÷ 2 = 12. (Actually, the "smallest" part is not as important as the "prime" part; the "smallest" part is mostly to make your work easier, because dividing by smaller numbers is simpler.) Now divide out the smallest number that goes into 12: 12 ÷ 2 = 6. Now divide out the smallest number that goes into 6: 6 ÷ 2 = 3. Since 3 is prime, you're done factoring, and the prime factorization is 2×2×2×3.
An easy way of keeping track of the factorization is to do upside-down division. It looks like this:
The nice thing about this upside-down division is that, when you're done, the prime factorization is the product of all the numbers around the outside. The factors are circled in red above. By the way, this upside-down division is something that should probably be done on scratch-paper, and not handed in as part of your homework. Copyright © Elizabeth Stapel 1999-2009 All Rights Reserved
* Find the prime factorization of 1050.
I'll do the upside-down division:
Then my answer is: 1050 = 2 × 3 × 5 × 5 × 7
Some texts prefer that answers such as this be written using exponential notation, in which case the final answer would be written as 2×3×52×7.
You can do the repeated division "right-side up", too, if you prefer. The process works the same way, but the division is reversed in orientation. The above problem would be worked out like this:
1050 = 2 x 3 x 5 x 5 x 7
*
Find the prime factorization of 1092.
I'll do the repeated division:
1092 = 2 x 2 x 3 x 7 x 13
1092 = 2 × 2 × 3 × 7 × 13
This answer might also be written as 22×3×7×13.
By the way, there are some divisibility rules that can help you find the numbers to divide by. There are many divisibility rules, but the simplest to use are these:
* If the number is even, then it's divisible by 2.
* If the number's digits sum to a number that's divisible by 3, then the number itself is divisible by 3.
* If the number ends with a 0 or a 5, then it's divisible by 5.
Of course, if the number is divisible twice by 2, then it's divisible by 4; if it's divisible by 2 and by 3, then it's divisible by 6; and if it's divisible twice by 3 (or if the sum of the digits is divisible by 9), then it's divisible by 9. But since you're finding the prime factorization, you don't really care about these non-prime divisibility rules. There is a rule for divisibility by 7, but it's complicated enough that it's probably easier to just do the division on your calculator and see if it comes out even.
If you run out of small primes and you're not done factoring, then keep trying bigger and bigger primes (11, 13, 17, 19, 23, etc) until you find something that works — or until you reach primes whose squares are bigger than what you're dividing into. Why? If your prime doesn't divide in, then the only potential divisors are bigger primes. Since the square of your prime is bigger than the number, then a bigger prime must have as its remainder a smaller number than your prime. The only smaller number left, since all the smaller primes have been eliminated, is 1. So the number left must be prime, and you're done.
Reference:
http://en.wikipedia.org/wiki/Factorization
http://www.purplemath.com/modules/factnumb.htm
No comments:
Post a Comment