Remainder and Factor Theorems

When a polynomial is divided by a linear polynomial, the remainder theorem is used to find the remainder.

A step-by-step guide to the remainder and factor theorems

According to the remainder theorem, if we divide a polynomial \(P(x)\) by the factor \((x – a)\); which is essentially not an element of a polynomial, you will find a smaller polynomial with the remainder. This remainder obtained is actually a value of \(P(x)\) at \(x = a\), specifically \(P(a)\). So basically, \((x -a)\) is the divisor of \(P(x)\) if and only if \(P(a) = 0\). It is applied to factorize polynomials of each degree elegantly.

The factor theorem states that if \(f(x)\) is a polynomial of degree \(n\) greater than or equal to \(1\), and \(a\) is any real number, then \((x – a)\) is a factor of \(f(x)\) if \(f(a) = 0\). In other words, we can say that \((x – a)\) is a factor of \(f(x)\) if \(f(a) = 0\).

Difference between the factor theorem and the remainder theorem

The remainder and factor theorems are similar but refer to two different concepts. The remainder theorem relates the remainder of the division of a polynomial by a binomial with the value of a function at a point. The factor theorem relates the factors of a given polynomial to its zeros.