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Students regularly ask questions about how to factor. Here are some answers from our archives; for the full answers, follow the links:

Introduction to Factoring
Could you explain the concept of factoring and give examples of its practical uses?
  Factoring is an idea you might be familiar with from multiplication. Numbers that can be multiplied together to get another number are its factors. For example, 4*3 = 12, so 3 and 4 are factors of 12. However, they're not its only factors; 1, 2, 6, and 12 are other factors of 12. (Another way of defining a factor is a number that goes evenly into the number you're factoring.)...

Finding All the Factors of a Number
How can I find the factors of 1-100?
  Do you know what a prime number is? A number is prime if it cannot be divided evenly by anything except itself and 1. For example, 5 is a prime number, because the only factors of 5 are 1*5=5. However, 12 is not a prime number, because 1*12=12, 2*6=12, and 3*4=12. Why should you care about prime numbers? For one thing, they make it easy to find the factors of a number...

Factoring Trinomials
I don't understand how to start out factoring this algebra.
  When you factor a number, say 12, you break it into its smallest parts. You could start with 6 x 2 and then 6 can be factored into 2 x 3, so the factors of 12 could be 2 x 2 x 3. To factor x^2 + 5x + 4 is to break it down into simpler parts.

Finding the Greatest Common Factor (GCF)
I am having trouble finding the greatest common factor (GCF) with exponents in the problems along with variables.
  Keep in mind that the GCF is the greatest common factor; that is, it is the biggest thing that is a factor of both expressions.

Finding LCDs, LCMs, and GCFs
I don't know how to do the greatest common factor, the least common multiple, and the least common denominator.
  Let's focus on what you do know, and maybe we'll discover that you know more than you think you do. Do you know how to express a number as the product of its prime factors? For example, do you know that 120 can be expressed (using x to indicate multiplication) as 2 x 2 x 2 x 3 x 5? ...

Factoring Fractions: Dividing Polynomials by Monomials
(12ax + 16x) / 4x = ? I just don't understand how to do this.
  Try factoring the first polynomial. Start with 12ax + 16x. The first thing I notice is that both expressions have an 'x' in them...

Factoring Quadratics
(2/y-1 + 6y-2) / (y^2+2y-3). I'm totally lost...
  To factor a polynomial there are basically three steps: 1. Identify and factor out common factors of all the coefficients. 2. Identify and factor out variables appearing in all the terms. 3. Factor the remaining piece, if possible....

Factoring Quadratic Equations
Can you explain in a simple form, how to do factoring? Following are some examples of expressions I need to factor.
  The idea in factoring is to look for a number or a variable that is present in every term of what needs factoring....

Factoring Quadratic Trinomials
I'm trying to figure out how to factor y^2 + 8y + 16. How do you know how to find the greatest common factor and factor it?
  In general, you will be trying to factor quadratic trinomials that look like a*y^2 + b*y + c where a, b, and c are expressions not involving y...

Solving Quadratic Equations by Factoring
I am very confused on how to do these problems.
  Step-by-step solutions for five different equations.

Factoring Polynomials
I'm having trouble factoring things like 12x^3y^9 + 20x^5y^4.
  Let's break the constants into prime factors, and expand out the exponents...

Polynomial Factoring Rules
I have a problem with the "special cases" of factoring poynomials.
  Basic rules for a^2-b^2, a^3-b^3, and a^3+b^3.

Factoring Polynomials Using Substitution
I can't get two questions on my own...
  Factoring x^2+10xy+25y^2-81z^2 and a^4-6a^2b^2-27b^4.

Difference between Two Cubes
Factoring a^3 - b^3.
  Looking at a^2 - b^2, you can see that it will be zero if a = b. If you have seen anything about solving quadratic equations by factoring, you will recognize that this means that a^2 - b^2 can be factored...

Factorizing Quadratic Trinomials
It says "factor as the product of linear binomials or write prime." Is there some trick to figuring this out?
  A way that works every time is to use the Quadratic Formula and then multiply through by the leading coefficient....

Completing the Square
I am taking my last 3 hours before I graduate in a three-week course. I have not taken any previous math...
  When you are faced with a quadratic polynomial in x, you can always write it as a constant times a square plus another constant, as follows....

Factoring Polynomials Using Logic
I keep trying to think of some logical way to deduce the answers to the problems, but I can't. The whole concept is fuzzy because, unlike so many other things in algebra that have definite formulas that are logical and clean-cut, factoring isn't this way.
  Suppose you want to factor a polynomial, say x^4 + 13x^2 + 36. You might try to factor it into two polynomials of degree 2. Now to determine a polynomial of degree 2, it is enough to find its value for three values of x, so let's substitute three values of x into the polynomial we're trying to factor....

More examples:
  Factoring a Trinomial
Factoring Trinomials - Difference of Two Squares
Quadratic Equations
Solving Equations by Factoring
General solutions to cubic and quartic polynomial equations

Some mathematical thought stemming from factoring: why, in a real equation, can't you have one real root and one imaginary root?
  Roots of the Quadratic Equation
Quadratic Roots
And see the Middle School Factoring section of the Dr. Math archives.

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