Lets take x= -1 
X^3 + x^2 -4x -4 
=(-1)^3 + (-1)^2 -4 (-1) -4 
= -1 +1 +4 -4 
=0..

Factors Theorem and Polynomials

Factors theorem is a concept in mathematics that helps us find factors of a polynomial expression. It states that if a polynomial P(x) has a factor (x - a), then P(a) = 0. This theorem is based on the fact that if (x - a) is a factor of P(x), then when we substitute 'a' in place of 'x', the value of the polynomial becomes zero.

Understanding Factors Theorem with an Example

Let's consider a polynomial P(x) = x^3 - 4x - 4. We can use the factors theorem to find factors of this polynomial by substituting different values of 'a' into P(x) and checking for which values P(a) = 0.

Using Factors Theorem to Find Factors

1. Let's check if (x - 1) is a factor of P(x). Substitute x = 1 in P(x): P(1) = 1^3 - 4(1) - 4 = 1 - 4 - 4 = -7, which is not equal to zero. Hence, (x - 1) is not a factor.

2. Now, let's check if (x + 1) is a factor of P(x). Substitute x = -1 in P(x): P(-1) = (-1)^3 - 4(-1) - 4 = -1 + 4 - 4 = -1, which is not equal to zero. Hence, (x + 1) is not a factor.

3. By using synthetic division or long division, we can find that (x + 1) is a factor of P(x). Dividing P(x) by (x + 1), we get the quotient as x^2 - x - 4.

This process helps us factorize polynomials and find their roots efficiently using the factors theorem.