8.1 THE FACTOR AND REMAINDER THEOREMS

If an nth degree polynomial p(x) = a₀xⁿ + a₁x^{n -1} + ... + a_n can be factored in the form
 

p(x) = a₀(x - r₁)(x - r₂)...(x - r_n), 

then the roots of the polynomial equation p(x) = 0 are found by setting each of the factors equal to zero, since a product of complex numbers is zero if and only if at least one of the factors is zero. Therefore, the roots are r₁, ..., r_n. We wish to establish a form of converse to this result: we wish to show that if r is a root of a polynomial equation p(x) = 0 then it follows that x - r is a factor of p(x); that is p(x) can be expressed in the form
 

where q(x) is a polynomial in x.

THE FACTOR THEOREM: Let

be a polynomial in x. If r is a root of p(x), that is, if p(r) = 0, then x - r is a factor of p(x).

Proof: Using the hypothesis that p(r) = 0, we have

Since x - r is a factor of xⁿ - rⁿ, x^{n -1} - r^{n -1}, and so on (see Example 2, Chapter 5), it follows that x - r is a factor of the entire right side of equation (1), and so is a factor of p(x).
 

Tuesday, June 2, 1998