Factor Theorem
Pronunciation: /ˈfæk tər ˈθɪər əm/ ?
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 2. |
References
- Wells, Webster. Factoring, pp 26. D. C. Heath & Co., Publishers, 1902. (Accessed: 2010-02-04). http://www.archive.org/stream/factoring00wellrich#page/26/mode/1up/search/theorem.
- Albert, A. Adrian. Introduction To Algebraic Theories, pp 4-6. The University of Chicago Press, 1941. (Accessed: 2010-02-04). http://www.archive.org/stream/introductiontoal033028mbp#page/n13/mode/1up/search/factor+theorem.
- Schultze, Arthur. Advanced Algebra, pp 8-17. The Macmillan Company, 1906. (Accessed: 2010-02-03). http://www.archive.org/stream/advancedalgebra00schugoog#page/n24/mode/1up/search/factor.
More Information
- Stapel, Elizabeth. The Factor Theorem. Purplemath. 2009-03-12. http://www.purplemath.com/modules/factrthm.htm.
Printed Resources
Cite this article as:
Factor Theorem. 2010-02-04. All Math Words Encyclopedia. Life is a Story Problem LLC. http://www.allmathwords.org/en/f/factortheorem.html.
Translations
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Revision History
2010-02-04: Added "References" (McAdams, David.)
2009-01-09: Initial version (McAdams, David.)