factorization
An expression listing items that, when multiplied together, will produce a desired quantity.
Examples of factorization in the following topics:

Greatest Common Factor and Factoring by Grouping
 Identify the greatest common factor of a list of integers and terms Breakdown a polynomial using the factor by grouping method Factoring by grouping divides the terms in a polynomial into groups, which can be factored using the greatest common factor.
 Factorization or factoring is the decomposition of an object, for example, a number or a polynomial, into a product of other objects, or factors, which when multiplied together give the original.
 Factor out the greatest common factor, 4x(x+5) + 3y(x+5).
 Factor out binomial (x+5)(4x+3y) Factorization or factoring is the decomposition of an object, for example, a number or a polynomial, into a product of other objects, or factors, which when multiplied together give the original.
 A way to factor some polynomials is factoring by grouping.
 factorization (noun) An expression listing items that, when multiplied together, will produce a desired quantity.

Finding Factors of Polynomials
 Breakdown a polynomial of the form x2a2 into its factors Show how to factor a polynomial by reversing the FOIL process Practice factoring a polynomial by pulling out the greatest common factor Finding factors of polynomials is important, since it is always best to work with the simplest version of a polynomial.
 There are four types of factoring shown which are "pulling out" common factors, factoring perfect squares, the difference between two squares, and then how to factor when the other three techniques are not applicable.
 When factoring, things are pulled apart.
 The common factor is 3.
 When factoring 3 from 6x, 2x is left.
 factor (verb) To find all the factors of (a number or other mathematical object) (the objects that divide it evenly).
 common factor (noun) A value, variable or combination of the two that is common to all terms of a polynomial.

Finding Zeroes of Factored Polynomials
 Use the factored form of a polynomial to find its zeros The factored form of a polynomial reveals its zeros, which are defined as points where the function touches the x axis.
 The factored form of a polynomial can reveal where the function crosses the x axis.
 "
Consider the factored function:
$f(x)=(xa_1)(xa_2)...  Consider the function:
$f(x)=x^3+2x^25x6$ This can be rewritten in the factored form:$f(x)=(x+3)(x+1)(x2)$ Replacing x with a value that will make (x+3), (x+1), or (x2) will result in f(x) being equal to zero.  Graph of cubic function
$f(x)=x^32x^25x6$ in factored form of$f(x)=(x+1)(x+3)(x2)$ .

Special Factorizations and Binomials
 Use special factoring techniques to factor binomials Differentiate between perfect square trinomials and nonperfect square trinomials By simplifying the FOIL method, you can save a lot of time when factoring for the difference of two square and perfect square trinomials.
 Perfect square trinomials always factor as the square of a binomial.
 Factor
$x^216$ .  Factor
$x^2+6x+9$ .  In the text, we outlined three steps to factoring out a trinomial.

Trinomials of the Form ax^2 + bx + c, Where a is Not Equal to 1
 Factor
$x^2  4x  21$ .  The factors are 9 and 8.
 Include these factors in the parentheses.
 Factor each parentheses.
 We are left with the proper factorization.
 Factor

Factoring Trinomials of the Form ax^2 + bx + c; Perfect Squares
 Differentiate between perfect square trinomials and nonperfect square trinomials
Apply factoring techniques to factor a trinomial of the form ax2 + bx + c
The polynomial
$ax^2 + bx + c$ can be factored using a variety of methods, including trial and error.  Some trinomials, known as perfect square trinomials, can be factored into two equal binomials.
 We can factor
$a^2 b^2$ , the difference of two squares, by finding the terms that produce the perfect squares and substituting these quantities into the factorization form.  When using real numbers, there is no factored form for the sum of two squares.
 Perfect square trinomials factor as the square of a binomial.
 Differentiate between perfect square trinomials and nonperfect square trinomials
Apply factoring techniques to factor a trinomial of the form ax2 + bx + c
The polynomial

Adding and Subtracting with Like and Unlike Denominators
 Always factor rational expressions before doing anything else.
 We start, as usual, by factoring.
 Similarly, the prime factors of 30 are 2×3×5.
 It is best to leave the bottom alone, since it is factored.
 But finally, we note that we can factor the top again.
 prime factor (noun) A factor of a given integer which is also a prime number.

Solving Quadratic Equations by Factoring
 For example, x^{2}  7x + 12 = 0 can be factored as (x  3)(x  4) = 0 so x  3 = 0 or x  4 = 0, yielding x = 3 or x = 4.
 To solve quadratic equations by factoring, lets remember the FOIL property.
 Before we jump into factoring out a quadratic equation, lets try to FOIL a function:
$(x3)(x4)=1x^24x3x+12=1x^27x+12=0$ Remembering this, lets think about how we need to go about factoring out a quadratic equation.  So if our last term has a positive sign, then both of the signs in our factored equation are the same.
 When factored out, this returns the original equation, so we know it is correct.
 factor (verb) To find all the factors of (a number or other mathematical object) (the objects that divide it evenly).

Logarithms of Products
 Apply the product rule for logarithms A useful property of logarithms states that the sum of two logarithms of factors is equal to the logarithm of the factors' product.
 The logarithm of a product is the sum of the logarithms of the factors.
 Tedious multidigit multiplication steps can be replaced by table lookups and simpler addition, because of the fact that the logarithm of a product is the sum of the logarithms of the factors:
$logb(xy) = logb(x) + logb(y)$ Of course, the reverse is true also.  The sum of two logarithms of factors is equal to the logarithm of the factors' product.

Integer Coefficients and the Rational Zeroes Theorem
 If a_{0} and a_{n} are nonzero, then each rational solution x, when written as a fraction x = p/q in lowest terms (i.e., the greatest common divisor of p and q is 1), satisfies 1) p is an integer factor of the constant term a_{0}, and 2) q is an integer factor of the leading coefficient a_{n}.
 A proof can be derived by first moving the constants to one side, factoring and multiplying by q^{n}.
 q is an integer factor of the leading coefficient a_{n}.
 +a_1(\frac pq)+a_0=0$
If we shift the constant term to the right hand side, factor a p and multiply by q^{n}, we get:
$p(a_np^{n1}+a_{n1}qp^{n2}+...  But p is coprime to q and therefore to q^{n}, so by (the generalized form of) Euclid's lemma, or first theorem, it must divide the remaining factor a0 of the product.
 Euclid's lemma (noun) In number theory, Euclid's lemma (also called Euclid's first theorem) is a lemma that captures one of the fundamental properties of prime numbers. It states that if a prime divides the product of two numbers, it must divide at least one of the factors. For example since 133 × 143 = 19019 is divisible by 19, one or both of 133 or 143 must be as well. In fact, 19 × 7 = 133. It is used in the proof of the fundamental theorem of arithmetic.
 coprime (adjective) Having no positive integer factors, aside from 1, in common with one or more specified other positive integers.