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 not a root of the polynomial, and (x-3) is a factor of the polynomial. |
Example 3. |
References
- Wells, Webster. Factoring. pp 26. www.archive.org. D. C. Heath & Co., Publishers. 1902. Last Accessed 8/6/2018. 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 8/6/2018. 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 8/6/2018. 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. 7/11/2018. All Math Words Encyclopedia. Life is a Story Problem LLC. http://www.allmathwords.org/en/f/factortheorem.html.
Revision History
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.)