# The Factor and Remainder Theorems: An overview

If, like many of my students, you are interested in passing A-level maths and familiar with polynomial equations, then you may have heard of the factor theorem and the remainder theorem.

The factor theorem and the remainder theorem are used to find solutions to polynomial equations by factoring them into their individual components. The factor theorem explains how a polynomial is related to its factors, while the remainder theorem explains how the remainder of a polynomial is related to its factors.

In this blog post, I will be taking a look at both the factor theorem and remainder theorem in more detail so that you can understand how they work and when they should be used. You might also enjoy reading: ​​why do we use the letter X to represent the unknown in maths?

## What is the Factor Theorem?

The factor theorem states that if a polynomial is divided by a linear factor, then it will remain divisible by this same linear factor if it has a zero as its solution. In other words, if you have a polynomial equation that, when divided by (x-a), has zero as its solution, then x-a must be one of the factors of that polynomial equation.

Put another way, if x-a is not one of the factors for the polynomial equation, then there will be no solution where x=a.

If a polynomial p(x) becomes zero when the number (x – a) is substituted then (x – a) is a factor of p(x) (Source: Michigan State University)

In other words, if (x – a) is a factor of a polynomial in x, the result is zero when a is substituted for x in the polynomial.

## What is the Remainder Theorem?

The remainder theorem states that if a polynomial expression P(x) is divided by (x-a), then the remainder after division will always equal P(a). In other words, when dividing P(x) by (x-a), whatever value you get for P(a) will also be your remainder after division.

This means that if you know what value P(A) equals without actually having to divide it, then you can use this information to determine whether or not (X-A) is a factor of P(X).

## Is Remainder Theorem and Factor Theorem the Same?

The factor theorem and remainder theorem are closely linked concepts in mathematics which are often used together when solving complex equations involving polynomials. While the factor theorem enables us to understand how a given linear factor relates to an entire polynomial equation, the remainder theorem helps us figure out what amount remains after dividing such an equation by said linear factor.

In other words, the remainder theorem states that when a polynomial P(x) is divided by (x-a), where “a” is any real number, then P(a) gives us the remainder. In other words, P(a) tells us what amount will remain after we divide our polynomial equation by (x-a).

On the other hand, the factor theorem states that if a polynomial equation is divided by a linear factor (x-a), then the remainder of the division will be equal to zero if and only if (x-a) is a factor of the equation. This means that if you divide an equation by (x-a) and are left with zero as your remainder, then (x-a) must be one of its factors.

To put it another way, if (x-a) is a factor of an equation, then one of its solutions will always be x=a because when x=a, there will not be any remainder after dividing by (x-a). Therefore, if you know that one solution to an equation is x=a then you can use this information to determine whether or not (x-a) is a factor of the equation using the factor theorem.

### In Conclusion

The factor theorem and remainder theorem are two important mathematical concepts that are essential for solving polynomial equations efficiently.

Both allow us to determine whether or not certain factors are present in an equation without having to actually perform the division itself, thus saving time and effort.

By understanding these two theorems, I believe that students can gain insight into how polynomial equations work and develop their problem-solving skills further.

Altiné

I am Altiné. I am the guy behind mathodics.com. When I am not teaching math, you can find me reading, running, biking, or doing anything that allows me to enjoy nature's beauty. I hope you find what you are looking for while visiting mathodics.com.