# Remainder and Factor Theorems

The factor theorem is mainly used for factoring polynomials and finding \(n\) roots of polynomials. In this step-by-step guide, you learn more about the factor and remainder 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 in an elegant manner.

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.

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