Remainder and Factor Theorem

The polynomial division algorithm is as follows: suppose d(x) and p(x) are nonzero polynomials where the degree of p(x) is greater than or equal to the degree of d(x). Then there exist two unique polynomials, q(x) and r(x), such that $p(x)=d(x)q(x)+r(x)$, where either r(x) = 0 or the degree of r(x) is strictly less than the degree of d(x).

Remainder Theorem: Suppose p(x) is a polynomial of degree at least 1 and c is a real number. When p(x) is divided by x − c the remainder is p(c).

Proof: By the division algorithm, $p(x)=(x-c)q(x)+r$, where r must be a constant since d(x) = x - c has a degree of 1. p(x) = (x - c)q(x) + r must hold for all values of x, so we can set x = c and get that $p(c)=(c-c)q(x)+r=r$. Thus the remainder r = p(c).

Factor Theorem: Suppose p(x) is a nonzero polynomial. The real number c is a zero of p if and only if (x − c) is a factor of p(x).

By the division algorithm, x - c is a factor of p(x) if and only if r = 0. So, since p(c) = r when p(x) is divided by x - c, x - c is a factor of p(x) if and only if p(c) = 0; that is, if c is a zero of p(x).

Resources

• The Factor Theorem and Remainder Theorem, Mathematics LibreTexts
• Dividing Polynomials, Paul's Online Notes