Factor Theorem
In algebraic math, the factor theorem is a theorem that establishes a relationship between factors and zeros of a polynomial. The Factor theorem is a unique case consideration of the polynomial remainder theorem. Thus the factor theorem states that a polynomial has a factor if and only if: The polynomial x – M is a factor of the polynomial f(x) if and only if f (M) = 0. Factor theorem is frequently linked with the remainder theorem, therefore do not confuse both. However, to unlock the functionality of the actor theorem, you need to explore through the remainder theorem.
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What is a Factor?
You now already know about the remainder theorem. The other most crucial thing we must understand through our learning for the factor theorem is what really a “factor” is. Knowing exactly what a “factor” is not only crucial to better understand the factor theorem, in fact, to all of mathematics concepts. It is a term you will hear time and again as you head forward with your studies.
With respect to division, a factor is an expression that, when a further expression is divided by this factor, the remainder is equal to zero (0). In its simplest form, take into account the following: 5 is a factor of 20 because, when we divide 20 by 5, we obtain the whole number 4 and no remainder. Likewise, 3 is not a factor of 20 because, when we 20 divided by 3, we have 6.67, which is not a whole number.
Use of Factor Theorem to Find the Factors of a Polynomial
In practical terms, the Factor Theorem is applied to factor the polynomials “completely”. In other words, any time you do the division by a number (being a prospective root of the polynomial) and obtain a remainder as zero (0) in the synthetic division, this indicates that the number is surely a root, and hence “x minus () the number” is a factor.
Now Before getting to know the Factor Theorem in depth and what it means, it is imperative that you completely understand the Remainder Theorem and what factors are first.
Remainder Theorem
As mentioned above, the remainder theorem and factor theorem are intricately related concepts in algebra. The reality is the former can’t exist without the latter and viceeversa. That being said, let’s see what the Remainder Theorem is.
By the principle of Remainder Theorem:

In case you divide a polynomial f(x) by (x – M), the remainder of that division is equal to f(c).
Usefulness of Remainder Theorem
The remainder theorem is particularly useful because it significantly decreases the amount of work and calculation that we would do to solve such types of mathematical problems/equations.
In absence of this theorem, we would have to face the complexity of using long division and/or synthetic division in order to have a solution for the remainder, which is both troublesome and time consuming. Moreover, an evaluation on the theories behind the remainder theorem, in addition to the visual proof of the theorem, is also quite useful.
Fun Facts

As per the Chaldean Numerology and the Pythagorean Numerology, the numerical value of factor theorem in is: 3

Factor theorem assures that a factor (x – M) for each root is r.

The factor theorem does not state there is only one such factor for each root. For the fact, it is quite easy to create polynomials with arbitrary repetitions of the same root & the same factor

Factor theorem is useful as it postulates that factoring a polynomial corresponds to finding roots.
Solved Examples
Problem 1:
Example: For a curve that crosses the xaxis at 3 points, of which one is at 2. What is the factor of 2x3−x2−7x+2?
Solution1:
The polynomial for the equation is degree 3, and could be all easy to solve. So let us arrange it first:
We can easily plot:
F (2) = 2 (2)3− (2)2− 7 (2) +2
This brings us:
= 16−4−14+2
=12 (14) +2
= 2 + 2
= 0
Thus! F (2) =0, so we have found a factor and a root
Therefore, (x2) should be a factor of 2x3−x2−7x+2
Problem 2:
Find out whether x + 1 is a factor of the below given polynomial.
i). 3 x 4 + x 3 – x2 + 3x + 2
Solution 2:
Using the factor theorem,
Let f(x) = 3×4 + x3 – x2 + 3x + 2
That brings to us:
f (–1) = 3 (–1) 4 + (–1) 3 – (–1)2 +3 (–1) + 2
= 3(1) + (–1) – 1 – 3 + 2 = 0
Hence, we conclude that (x + 1) is a factor of f (x)
Q.1. Is Factor Theorem and Remainder Theorem the Same?
These two theorems are not the same but dependent on each other. While the remainder theorem makes you aware of any polynomial f(x), if you do the division by the binomial x−M, the remainder is equivalent to the value of f (M). Whereas, the factor theorem makes aware that if a is a zero of a polynomial f(x), then (x−M) is a factor of f(M) , and viceversa.
Q.2. What is a Remainder Factor Theorem?
In terms of algebra, the remainder factor theorem is in reality two theorems that links the roots of a polynomial in accordance to its linear factors. The theorem is commonly used to easily help factorize polynomials while skipping the use of long or synthetic division. Particularly, when put in combination with the rational root theorem, this provides for a powerful tool to factor polynomials.
Q.3. What Does a Factor Theorem State?
By the rule of the Factor Theorem, if we do the division of a polynomial f(x) by (x – M), and (x – M) is a factor of the polynomial f(x), then the remainder of that division is equal to 0. Thus, as per this theorem, if the remainder of a division equals zero, (x – M) should be a factor. But, in case the remainder of such a division is NOT 0, then (x – M) is NOT a factor.