Factorization
In
mathematics, factorization or factoring is the decomposition of an object 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 . In all cases, a product of simpler things 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.
Encyclopedia
In
mathematics,
factorization or
factoring is the decomposition of an object 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 . In all cases, a product of simpler things 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.
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.
Prime factorization of an integer
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. For smaller numbers, however, there are a variety of different algorithms that can be applied.
- See also: Prime factorization algorithms
Factoring a quadratic polynomial
Any quadratic
polynomial can be factored into an expression with the form using the
quadratic formula. The method is as follows:
where
j and
k are the two
roots of the polynomial, found with the
quadratic formula.
Quadratic polynomials can sometimes be factored into two binomials with simple integer coefficients, without the need to use the quadratic formula. In a
quadratic equation, this will expose its two roots. The formula
would be factored into:
where
-
-
-
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 2
x2 − 5
x + 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 = 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 tells us that 2
x2 − 5
x + 2 = 0 factors into = 0, giving us the roots
x =
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:
Table method
The
table method is less used method that involves creating a multiplication table. This method is very decisive and can be much faster than other algorithms once the rules surrounding the method are mastered.
To factor the polynomial
ax2 +
bx +
c:
- place the value of a in the top-left of the table and the last term in the bottom-right box,
- find two factors of a × c that add up to b and place those factors in the top-right and bottom-left boxes,
- find the greatest common factor of each adjacent pair
The table will then be in this form:
| a | j | | A = GCF |
| k | c | | B = GCF |
| |
| C = GCF | D = GCF | | |
where
j and
k are factors of and
j+
k=
b, and
A,
B,
C, and
D are lables for the GCFs.
The GCFs comprise the final factors of the quadratic in this way:
- one factor is if k is positive, or if k is negative, and
- the other factor is if j is positive, or if j is negative.
Example
For example, to factor 6
x2 − 17
x + 12, first multiply 6
x2 × 12 to get 72
x2. Then find two factors of 72 that add up to -17 .
In the table, place the
a value in the top-left and the last term in the bottom-right box. Fit the -9 and -8 in the remaining boxes. Then find the GCF up and down and side to side for each row for the answer.
The final answer is .
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, can be factored into .
Sum/difference of two cubes
Another less-used but still common formula for factoring is the sum or difference of two cubes. The sum can be represented by
and the difference by
For example,
x3 − 10
3 can be factored into .
In general, the difference of any two numbers raised to the same power
contains the factor , and the sum of any two numbers raised to the same odd power
and the difference of any two numbers raised to the same even power contain the factor .
Factoring by grouping
Another way to factor some equations is factoring by grouping. This is done by placing the terms in an expression 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, suppose you had the expression
which upon first glance looks like a crazy expression. One logical step, if you decide to factor by grouping, would be to combine all of the expressions with sin
x and all without sin
x. Then you would have the expression
where each of the two groups can be factored giving us
This can be further simplified into
when can then be factored into
which is the expression in fully factored form.
Factoring in mathematical logic
In mathematical logic and automated theorem proving, factoring is the technique of deriving a single, more specific
atom from a disjunction of two more general unifiable atoms. For example, from ?
X,
Y :
P or
P we can derive
P.
See also
- Integer factorization
- Prime factorization algorithm
- Program synthesis
- Matrix decomposition
- Unique factorization
- Polynomial expansion, the opposite of factorization
- Factor group
- Factor ring
External links
-
- is an online factorization tool.
- is a comprehensive tutorial resource on basic factoring of polynomials.
