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,
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
- Is 2 a root of
x^{3} + 2x^{2} - 5x - 6?
Step | Equation | Description |
1 |
x^{3} + 2x^{2} - 5x - 6 ≟ 0, x = 2 |
This is the case to test |
2 |
2^{3} + 2 · 2^{2} - 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. |
- Is 1 a root of
x^{3} + 2x^{2} - 5x - 6?
Step | Equation | Description |
1 | x^{3} + 2x^{2}
- 5x - 6 ≟ 0, x = 1 |
This is the case to test |
2 | 1^{3} + 2 · 1^{2} - 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. |
- Is (x - 3) a factor of
x^{3} - 7x - 6?
Step | Equation | Description |
1 | x^{3} - 7x - 6 ≟ 0, x = 3 |
This is the case to test |
2 | 3^{3} - 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
- 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
- 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
- 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
- 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.)