factor theorem


Definition of Factor Theorem

  • According to Factor Theorem, if s is the root of any polynomial P(x), i.e. if P(s) = 0, then (x - s) is the factor of the polynomial P(x).

Example of Factor Theorem

  • To factor x3 + 3x2 + 3x + 1, first find the zero of the polynomial.
    Let f(x) = x3 + 3x2 + 3x + 1
    Let us check x = 1 by trial and error method, then, f(1) = (1)3 + 3(1)2 + 3(1) + 1 = 1 + 3 + 3 + 1 = 8 which is not equal to zero.
    So, (x – 1) is not a factor.
    Now check x = - 1, f(- 1) = (- 1)3 + 3(- 1)2 + 3(- 1) + 1 = - 1 + 3 - 3 + 1 = 0.
    Therefore, (x + 1) is a factor of the given polynomial.
    By dividing the given polynomial with (x + 1), we get (x2 + 2x + 1).
    Factoring this trinomial, we get (x + 1) (x + 1).
    So, the factors of the given polynomial are (x + 1), (x + 1), and (x + 1).

Solved Example on Factor Theorem

Identify the factors of the polynomial 2x2 - 5x + 3.
Choices:
A. (2x - 3)(x - 1)(x + 1)
B. (2x - 3)(x - 1)
C. (2x - 3)(x - 2)(x + 1)
D. (2x - 3)(x + 1)(x + 2)
Correct Answer: B
Solution:
Step 1: Let f(x) = 2x2 - 5x + 3
Step 2: Let us check x = 1 by trial and error method. Then, f(1) = 2(1)2 - 5(1) + 3 = 2 - 5 + 3 = 0
Step 3: According to factor theorem, if f(1) = 0, then (x - 1) is a factor of the polynomial f(x).
Step 4: Therefore, (x - 1) is a factor of the given polynomial.
Step 5: By dividing the given polynomial with (x - 1), we get (2x - 3)
Step 6: So, the factors of the given polynomial are (2x - 3) and (x - 1).

Related Terms for Factor Theorem

  • Factor
  • Polynomial