[Maths Class Notes] on Factor Theorem Pdf for Exam

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

 

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) = 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)