Factoring Polynomials
Pronunciation: /ˈfæk tər ɪŋ ˌpɒl əˈnoʊ mi əlz/ ?

Figure 1: A factored polynomial. 

To factor a polynomial is to find two or more
factors
of a
polynomial.
The factors of a polynomial are a set of polynomials of lesser or
equal degree that, when
multiplied
together, make the original polynomial. To
factor a polynomial completely is to find the factors of least
degree
that, when multiplied together, make the original polynomial.
Stated mathematically, to factor a polynomial P(x), is to find two or
more polynomials, say Q(x) and R(x), of lesser degree such that
P(x) = Q(x) · R(x). In example 1, x3, 2x+1,
and (x+2) are factors of the polynomial
2x^{3}  x^{2}  13x  6.

Factoring Step 1: Factor Out the Greatest Common Factor
To factor a polynomial, first find the
greatest common factor
of the terms. The greatest common factor can be found by:
 Finding the prime factorization
of each of the terms.
 Identifying the common factors.
 Multiplying the common factors together to get the greatest common factor.
Step  Figure  Description 
1   This is the polynomial to factor. 
2  Find the greatest common factor of the terms. 
2.1   Find the prime factorization of the terms. The prime factors of 2x are 2 and x. The prime factors of 4 are 2 and 2. The 1 is used to preserve the sign. Since the value of x is unknown, for the purposes of factoring, it is taken to be prime. 
2.2   Identify the common factors. In this case, 2 is the only common factor. 
2.3   Calculate the greatest common factor (GCF). Since the only common factor is 2, the greatest common factor must also be 2. 
3   Rewrite the polynomial as the greatest common factor multiplied by each of the terms with the greatest common factor removed. 
4   Check the work by multiplying the polynomials using the distributive property of multiplication. 
Example 1 
Step  Figure  Description 
1   This is the polynomial to factor. 
2  Find the greatest common factor of the terms. 
2.1   Find the prime factorization of the terms. The prime factors of 3x^{2} are 3·x·x . The prime factor of x is x. The prime factor of 2 is 2. Since the value of x is unknown, for the purposes of factoring, it is taken to be prime. 
2.2   Identify the common factors. In this case, 1 is the only common factor. Since a factor of 1 is not useful, this step is done. 
Example 2 
Step  Figure  Description 
1   This is the polynomial to factor. 
2  Find the greatest common factor of the terms. 
2.1   Find the prime factorization of the terms. The prime factors of 14x^{4} are 2·7·x·x·x·x. The prime factors of 2x^{2} are 2·x·x. The 1 is used to preserve the sign. Since the value of x is unknown, for the purposes of factoring, it is taken to be prime. 
2.2   Identify the common factors. In this case, 2, x and x are the common prime factors. 
2.3   Calculate the greatest common factor (GCF). Since the common prime factors are 2, x and x, the greatest common factor is 2·x·x = 2x^{2}. 
3   Rewrite the polynomial as the greatest common factor multiplied by each of the terms with the greatest common factor removed. 
4   Check the work by multiplying the polynomials using the distributive property of multiplication. 
Example 3 
Factoring Polynomials of Degree 2 Using the Quadratic Formula
The second step in factoring a
degree
2 polynomial, also called a
quadratic equation,
is to determine if it can be factored using real numbers. The
discriminant
of a degree 2 polynomial tells how many and what types of solutions can be
found for a quadratic equation. For a quadratic equation
f(x) = ax^{2}+bx+c, the discriminant is b^{2}  4ac.
Discriminant  Solutions  Example 
b^{2}  4ac < 0  Two complex solutions; No real solutions.  
b^{2}  4ac = 0  One real solution.  
b^{2}  4ac > 0  Two real solutions.  
Table 1: Discriminant and solutions of a quadratic equation  
If the are no real solutions, the quadratic can not be factored further without
using complex numbers. If there are real solutions, use the quadratic function to
find the solutions.
Factoring a Quadratic With Two Real Solutions
Equation  Description 
 This is the equation to factor. 
 Since we are finding the roots of the polynomial, set y to 0. 
 Identify a, b, and c in the quadratic function. a = 1 and b = 1 since the implied coefficients of x^{2} and x are 1. 
 Fill in the quadratic formula. 
 Simplify the exponent. 
 Simplify multiplication. 
 Simplify addition and subtraction. 
 Simplify the square root. 
 Split the solution into two explicit equations. 
 Simplify the numerators. 
 Simplify the fractions. 
 Get zero on one side of the equation to make the factors. 
 Rewrite the equation using the factors. 
 Check the work. Use the FOIL method to multiply out the factors. 
 Simplify the terms and add like terms. This is the original equation, so the factoring is correct. 
Example 4. 
Factoring a Quadratic With One Real Solution
Equation  Description 
 This is the equation to factor. 
 Since we are finding the roots of the polynomial, set y to 0. 
 Identify a, b, and c in the quadratic function. a = 1 since the implied coefficient of x^{2} is 1. 
 Fill in the quadratic formula. 
 Simplify the exponent. 
 Simplify multiplication. 
 Simplify addition and subtraction. 
 Since , substitute for . 
 Since anything plus zero is itself and anything minus zero is itself, the plus or minus zero goes away. 
 Reduce the fraction. 
 Subtract 1 from both sides to get the factor. 
 Write out the factors. Since there is only one root of the polynomial, it is a double root. 
 Check the work. Use the FOIL method to multiply out the factors. 
 Simplify the terms and add like terms. This is the original equation, so the factoring is correct. 
Example 5. 
Factoring a Quadratic With No Real Solutions
If the range of the equation is limited to real numbers, a quadratic with no real
solutions can not be factored any further. Example 4 shows how to solve a quadratic
equation with no real solutions.
Equation  Description 
 This is the equation to factor. 
 Since we are finding the roots of the polynomial, set y to 0. 
 Identify a, b, and c in the quadratic function. a = 1 and b = 1 since the implied coefficients of x^{2} and x are 1. 
 Fill in the quadratic formula. 
 Simplify the exponent. 
 Simplify multiplication. 
 Simplify addition and subtraction. 
 Since , substitute for . 
 Split the solution into two explicit equations. 
 Get 0 on one side of the equation to get the factors. 
 Rewrite the equation using the factors. 
 Check the work. Use the FOIL method to multiply out the factors. 
 Simplify the terms and add like terms. This is the original equation, so the factoring is correct. 
Example 6. 
Factoring Polynomials With Degree 3 or Greater
The formulas for solving polynomials with degree 3 and degree 4 are so complicated, that
they are only useful in computer programs. Polynomials of degree 5 and higher can
not be solved with a formula. The
fundamental theorem of algebra
implies that every nonconstant polynomial with real coefficients can be factored
over the real numbers into a product of linear factors and irreducible quadratic
factors. While the fundamental theorem of algebra says the factors exist, it does
not tell us how to find them. There are many methods that can be used. The
rational roots theorem
can be used to find possible roots of a polynomial.
Interpolation
is another method that can be used to find the roots of a polynomial.
Graphical solutions
can also approximate roots and factors of a polynomial.
References
 Wells, Webster. Factoring. D. C. Heath & Co., Publishers, 1902. (Accessed: 20100204). http://www.archive.org/stream/factoring00wellrich#page/26/mode/1up/search/theorem.
 Albert, A. Adrian. Introduction To Algebraic Theories, pp 118. The University of Chicago Press, 1941. (Accessed: 20100204). http://www.archive.org/stream/introductiontoal033028mbp#page/n13/mode/1up/search/factor+theorem.
 Bettinger, Alvin K. and Englund, John A.. Algebra and Trigonometry, pp 2530. International Textbook Company, January 1963. (Accessed: 20100112). http://www.archive.org/stream/algebraandtrigon033520mbp#page/n18/mode/1up.
 E.J. Barbeau. Polynomials, pp 93104. Springer, October 9, 2003.
More Information
 McAdams, David. Solving Graphically. allmathwords.org. All Math Words Encyclopedia. Life is a Story Problem LLC. 20090312. http://www.allmathwords.org/article.aspx?lang=en&id=Solving Graphically.
 McAdams, David. Interpolation. allmathwords.org. All Math Words Encyclopedia. Life is a Story Problem LLC. 20090312. http://www.allmathwords.org/article.aspx?lang=en&id=Interpolation.
 McAdams, David. Rational Roots Theorem. allmathwords.org. All Math Words Encyclopedia. Life is a Story Problem LLC. 20090312. http://www.allmathwords.org/article.aspx?lang=en&id=Rational%20Roots%20Theorem.
 How to Use UnFOIL to Factor Quadratic Equations (video). dummies.com. Wiley. 20100123. http://www.dummies.com/howto/content/howtouseunfoiltofactorquadraticequations.html.
Printed Resources
Cite this article as:
Factoring Polynomials. 20100204. All Math Words Encyclopedia. Life is a Story Problem LLC. http://www.allmathwords.org/en/f/factoringpolynomials.html.
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20100204: Added "References" (
McAdams, David.)
20090504: Corrected text for example 2. (
McAdams, David.)
20090120: Initial version (
McAdams, David.)