# Factor Theorem

Pronunciation: /ˈfæk.tər ˈθɪər.əm/ Explain

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

(x - a) is a factor of P(x) if and only if P(a) = 0.
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, 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?

StepEquationDescription
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?

StepEquationDescription
1x3 + 2x2 - 5x - 6 ≟ 0, x = 1 This is the case to test
213 + 2 · 12 - 5 · 1 - 6 ≟ 0 Substitute 1 in for x.
31 + 2 · 1 - 5 · 1 - 6 ≟ 0 Simplify the exponents.
41 + 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?

StepEquationDescription
1x3 - 7x - 6 ≟ 0, x = 3 This is the case to test
233 - 7 · 3 - 6 ≟ 0 Substitute 3 in for x.
327 - 7 · 3 - 6 ≟ 0 Simplify the exponents.
427 - 21 - 6 ≟ 0 Simplify the multiplication.
50 ≟ 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

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

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.)