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

  1. Is 2 a root of x3+2x2-5x-6?
    StepEquationDescription
    1x3+2x2-5x-6 ≟ 0, x=2This is the case to test
    223+2·22-5·2-6 ≟ 0Substitute 2 in for x.
    38+2·4-5·2-6 ≟ 0Simplify the exponents.
    48+8-10-6 ≟ 0Simplify the multiplication.
    50 ≟ 0Simplify 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=1This is the case to test
    213+2·12-5·1-6 ≟ 0Substitute 1 in for x.
    31+2·1-5·1-6 ≟ 0Simplify the exponents.
    41+2-5-6 ≟ 0Simplify the multiplication.
    5-8 ≟ 0Simplify 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=3This is the case to test
    233-7·3-6 ≟ 0Substitute 3 in for x.
    327-7·3-6 ≟ 0Simplify the exponents.
    427-21-6 ≟ 0Simplify the multiplication.
    50 ≟ 0Simplify 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

    1. 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.
    2. 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.
    3. 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.

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    2010-02-04: Added "References" (McAdams, David.)
    2009-01-09: Initial version (McAdams, David.)

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