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

Factors
 This is a complete list of the factors of 24.
 Therefore, 2 and 3 are prime factors of 6.
 However, 6 is not a prime factor.
 To factor larger numbers, it can be helpful to draw a factor tree.
 This factor tree shows the factorization of 864.

Introduction to Factoring Polynomials
 Factoring by grouping divides the terms in a polynomial into groups, which can be factored using the greatest common factor.
 Factor out the greatest common factor, $4x(x+5) + 3y(x+5)$.
 Factor out binomial $(x+5)(4x+3y)$.
 One way to factor polynomials is factoring by grouping.
 Both groups share the same factor $(x+5)$, so the polynomial is factored as

Factoring General Quadratics
 Polynomials of the form $ax^2+bx+c$ can be factored via the trial and error method.
 We can factor quadratic equations of the form $ax^2 + bx + c$ by first finding the factors of the constant $c$.
 The factored form of a quadratic equation takes the general form:
 This leads to the factored form:
 First, we factor $a$, which yields the set $3 \text{ & } 2$.

Factoring Perfect Square Trinomials
 When a trinomial is a perfect square, it can be factored into two equal binomials.
 It is important to be able to recognize such trinomials, so that they can the be factored as a perfect square.
 If you are attempting to to factor a trinomial and realize that it is a perfect square, the factoring becomes much easier to do.
 Since the middle term is twice $4 \cdot x$, this must be a perfect square trinomial, and we can factor it as: $x^2+8x+16=(x+4)^2.$
 Evaluate whether a quadratic equation is a perfect square and factor it accordingly if it is

Finding Factors of Polynomials
 When factoring, things are pulled apart.
 There are four basic types of factoring.
 The common factor is $3$.
 This is the simplest kind of factoring.
 Therefore it factors as $(x+5)(x5)$.

Solving Quadratic Equations by Factoring
 To factor an expression means to rewrite it so that it is the product of factors.
 The reverse process is called factoring.
 You may be wondering: but how does one come up with the two factors?
 Again, imagine you want to factor $x^27x+12$.
 We would need factors of $8$ which combine to be $2$.

Finding Zeroes of Factored Polynomials
 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.
 In general, we know from the remainder theorem that $a$ is a zero of $f(x)$ if and only if $xa$ divides $f(x).$ Thus if we can factor $f(x)$ in polynomials of as small a degree as possible, we know its zeros by looking at all linear terms in the factorization.
 This is why factorization is so important: to be able to recognize the zeros of a polynomial quickly.
 Use the factored form of a polynomial to find its zeros

Factoring a Difference of Squares
 When a quadratic is a difference of squares, there is a helpful formula for factoring it.
 For example, suppose you were trying to factor the quadratic expression $x^216.$ If you recognize the first term as the square of $x$ and the term after the minus sign as the square of $4$, you can then factor the expression as $(x4)(x+4).$
 By rewriting the equation as $x^2a^2=0$ and factoring the difference of squares, you would obtain $(xa)(x+a)=0.$ Thus there are two solutions, where $xa=0$ (so $x=a$) and where $x+a=0$, (so $x=a$).
 This latter equation has no solutions, since $4x^2$ is always greater than or equal to $0.$ However, the first equation $4x^23=0$ can be factored again as the difference of squares, if we consider $3$ as the square of $\sqrt3$.
 Evaluate whether a quadratic equation is a difference of squares and factor it accordingly if it is.

Rational Algebraic Expressions
 We start, as usual, by factoring.
 Similarly, the prime factors of 30 are 2, 3, and 5.
 This requires factoring algebraic expressions.
 Notice the factors in the denominators.
 The second fraction has one factor: $(x^2 + 2)$.

Rules for Exponent Arithmetic
 $a^m$ means that you have $m$ factors of $a$.
 If you multiply this quantity by $a^n$, i.e. by $n$ additional factors of $a$, then you have $a^{m+n}$ factors in total.
 In the same way that ${ a }^{ m }\cdot { a }^{ n }={ a }^{ m+n }$ because you are adding on factors of $a$, dividing removes factors of $a$.
 If you have $n$ factors of $a$ in the denominator, then you can cross out $n$ factors from the numerator.
 If there were $m$ factors in the numerator, now you have $(mn)$ factors in the numerator.