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).
Video Examples: Factor Theorem and The Remainder Theorem
Example of Factor Theorem

To factor x^{3} + 3x^{2} + 3x + 1, first find the zero of the polynomial.
Let f(x) = x^{3} + 3x^{2} + 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 (x^{2} + 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
Ques: Identify the factors of the polynomial 2x^{2 } 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) = 2x^{2}  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).
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