An Introduction to Factor
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 the remainder theorem.
Factor theorem is a theorem that helps to establish a relationship between the factors and the zeros of a polynomial. The Factor Theorem is said to be a unique case consideration of the polynomial remainder theorem. The factor theorem states that a polynomial has a factor provided the polynomial x - M is a factor of the polynomial f(x) island provided f f (M) = 0.
Factor theorem is frequently linked with the remainder theorem.
Factor
In division, a factor refers to an expression which, when a further expression is divided by this particular factor, the remainder is equal to zero (0). Let us take the following: 5 is a factor of 20 since, when we divide 20 by 5, we get the whole number 4 and there is no remainder. Similarly, 3 is not a factor of 20 since when we 20 divide by 3, we have 6.67, and this is not a whole number.
Remainder Theorem
According to the principle of Remainder Theorem:
If we divide a polynomial f(x) by (x - M), the remainder of that division is equal to f(c).
The Usefulness of Remainder Theorem
This Remainder theorem comes in useful since it significantly decreases the amount of work and calculation that could be involved to solve such problems/equations.
Without this Remainder theorem, it would have been difficult to use long division and/or synthetic division to have a solution for the remainder, which is difficult time-consuming.
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 a "factor" is. Knowing exactly what a "factor" is not only crucial to better understand the factor theorem, in fact, to all mathematics concepts. It is a term you will hear time and again as you head forward with your studies.
Concerning 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 are 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 vice-e-versa. 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).
The 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 to have a solution for the remainder, which is both troublesome and time-consuming. Moreover, an evaluation of 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 the factor theorem 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 this 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
1. Example: For a curve that crosses the x-axis at 3 points, of which one is at 2. What is the factor of 2x3−x2−7x+2?
Ans: 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, (x-2) should be a factor of 2x3−x2−7x+2.
2. Find out whether x + 1 is a factor of the below-given polynomial.
i) 3 x 4 + x 3 – x2 + 3x + 2
Ans: Using the factor theorem,
Let f(x) = 3x4 + 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)
FAQs on Factor Theorem
1. Is the factor Theorem and the Remainder Theorem the same?
These two theorems are not the same but both of them are dependent on each other. With the Remainder theorem, you get to know of any polynomial f(x), if you divide by the binomial x−M, the remainder is equivalent to the value of f (M). On the other hand, the Factor theorem makes us aware that if a is a zero of a polynomial f(x), then (x−M) is a factor of f(M), and vice-versa.
2. Explain the Remainder Factor Theorem?
In purely Algebraic terms, the Remainder factor theorem is a combination of two theorems that link the roots of a polynomial following its linear factors. This theorem is mainly used to easily help factorize polynomials without taking the help of the long or the synthetic division process. When it is put in combination with the rational root theorem, this theorem provides a powerful tool to factor polynomials.
3. What does the Factor theorem state?
According to the rule of the Factor Theorem, if we take the division of a polynomial f(x) by (x - M), and where (x - M) is a factor of the polynomial f(x), in that case, the remainder of that division will be equal to 0. Therefore, according to this theorem, if the remainder of a division is equal to zero, in that case,(x - M) should be a factor, whereas if the remainder of such a division is not 0, in that case,(x - M) will not be a factor.
4. Explain Factor Theorem.
Factor theorem is a polynomial remainder theorem that links the factors of a polynomial and its zeros together. In the factor theorem, all the known zeros are removed from a given polynomial equation and leave all the unknown zeros. The polynomial we get has a lower degree where the zeros can be easily found out.
It is very helpful while analyzing polynomial equations. Factoring comes in useful in real life too, while exchanging money, while dividing any quantity into equal pieces, in understanding time, and also in comparing prices.
5. Where can I get study notes on Algebra?
Maths is an all-important subject and it is necessary to be able to practice some of the important questions to be able to score well. The online portal, Vedantu.com offers important questions along with answers and other very helpful study material on ‘Factor Theorem’, which have been formulated in a well-structured, well researched, and easy to understand manner. These study materials and solutions are all important and are very easily accessible from Vedantu.com and can be downloaded for free.
6. 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 divide 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 vice-versa.
7. What is a Remainder Factor Theorem?
In terms of algebra, the remainder factor theorem is in reality two theorems that link the roots of a polynomial following 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.
8. 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.