Factor Theorem

Pronunciation: /ˈfæk.tər ˈθɪər.əm/ Explain

The factor theorem states that, for a number a and a polynomial P(x):

This says that x - a is a factor of P(x) if and only if a is a zero of P(x). If (x - a) is a factor of polynomial P(x), then there exists a polynomial P'(x) such that (x - a)P'(x) = P(x). Substitute a in for x to get (a - a)P'(a) = P(a). Since a - a = 0, 0·P'(a) = 0 = P(a).

The factor theorem can be used to find out if a number is a root of a polynomial. If the number is substituted into a polynomial, and the result is 0, then the number is a root of the polynomial.

Examples

1. Is 2 a root of x³ + 2x² - 5x - 6?

   Step	Equation	Description
   1	x3 + 2x2 - 5x - 6 ≟ 0, x = 2	This is the case to test
   2	23 + 2 · 22 - 5 · 2 - 6 ≟ 0	Substitute 2 in for x.
   3	8 + 2 · 4 - 5 · 2 - 6 ≟ 0	Simplify the exponents.
   4	8 + 8 - 10 - 6 ≟ 0	Simplify the multiplication.
   5	0 ≟ 0	Simplify the addition. Since 0 = 0, 2 is a root of the polynomial and (x - 2) is a factor of the polynomial.
   Example 1.

2. Is 1 a root of x³ + 2x² - 5x - 6?

   Step	Equation	Description
   1	x3 + 2x2 - 5x - 6 ≟ 0, x = 1	This is the case to test
   2	13 + 2 · 12 - 5 · 1 - 6 ≟ 0	Substitute 1 in for x.
   3	1 + 2 · 1 - 5 · 1 - 6 ≟ 0	Simplify the exponents.
   4	1 + 2 - 5 - 6 ≟ 0	Simplify the multiplication.
   5	-8 = 0	Simplify the addition. Since -8 ≠ 0, 1 is not a root of the polynomial and (x - 1) is not a factor of the polynomial.
   Example 2.

3. Is (x - 3) a factor of x³ - 7x - 6?

   Step	Equation	Description
   1	x3 - 7x - 6 ≟ 0, x = 3	This is the case to test
   2	33 - 7 · 3 - 6 ≟ 0	Substitute 3 in for x.
   3	27 - 7 · 3 - 6 ≟ 0	Simplify the exponents.
   4	27 - 21 - 6 ≟ 0	Simplify the multiplication.
   5	0 ≟ 0	Simplify the addition. Since 0 = 0, 3 is a root of the polynomial, and (x - 3) is a factor of the polynomial.
   Example 3.

   References

   1. McAdams, David E.. All Math Words Dictionary, factor theorem. 2nd Classroom edition 20150108-4799968. pg 78. Life is a Story Problem LLC. January 8, 2015. Buy the book
   2. Wells, Webster. Factoring. pp 26. www.archive.org. D. C. Heath & Co., Publishers. 1902. Last Accessed 7/11/2018. http://www.archive.org/stream/factoring00wellrich#page/26/mode/1up/search/theorem. Buy the book
   3. Albert, A. Adrian. Introduction To Algebraic Theories. pp 4-6. www.archive.org. The University of Chicago Press. 1941. Last Accessed 7/11/2018. http://www.archive.org/stream/introductiontoal033028mbp#page/n13/mode/1up/search/factor+theorem. Buy the book
   4. Schultze, Arthur. Advanced Algebra. pp 8-17. www.archive.org. The Macmillan Company. 1906. Last Accessed 7/11/2018. http://www.archive.org/stream/advancedalgebra00schugoog#page/n24/mode/1up/search/factor. Buy the book

   More Information

   • Stapel, Elizabeth. The Factor Theorem. Purplemath. 3/12/2009. http://www.purplemath.com/modules/factrthm.htm.

   Cite this article as:

   McAdams, David E. Factor Theorem. 4/20/2019. All Math Words Encyclopedia. Life is a Story Problem LLC. http://www.allmathwords.org/en/f/factortheorem.html.

   Revision History

   4/20/2019: Modified equations and expression to match the new format. (McAdams, David E.)
   12/21/2018: Reviewed and corrected IPA pronunication. (McAdams, David E.)
   7/9/2018: Removed broken links, updated license, implemented new markup, implemented new Geogebra protocol. (McAdams, David E.)
   2/4/2010: Added "References". (McAdams, David E.)
   1/9/2009: Initial version. (McAdams, David E.)