Factor Theorem

The factor theorem is particularly beneficial in polynomial divisions, graphing functions, and factoring polynomials completely.
The theorem suggests; if f(a) =0, for a polynomial f(x), then (x-a) is a factor of f(x).

The remainder theorem helps in finding remainders without going through the whole process of long division. It states that when a linear divisor (x-a) divides a polynomial f(x), the remainder is the value of the polynomial at x = a, i.e., f(a). The factor theorem is a special case of the remainder theorem

The factor theorem states:
For a polynomial f(x), if f(a) = 0, then (x - a) is a factor of f(x). 
Conversely, if (x - a) is a factor of f(x), then f(a) = 0
For instance, let f(x) = x³ -6x² + 11x - 6.
  Check f(1),
 f(1) = 1³ -  6(1)² + 11(1) - 6 = 0
 

So, by factor theorem, (x-1) is a factor of f(x)

Proof of Factor Theorem: 
According to division algorithm for polynomials:
f(x) = (x - a)  q(x) + r
Where:
f(x) is the original polynomial
(x - a) is the divisor
q(x) is the quotient polynomial
r is the remainder   

Substituting x=a
f(a)= (a - a)   q(a) + r =  r           
So, f(a) = r
If f(a) = 0, then r = 0
f(x) = (x-a)    q(x)
So,  
(x-a) is a factor of f(x)
The factor theorem is a result of the remainder theorem.

A value that, when substituted with the variable, makes the whole polynomial value zero is the zero of that polynomial. Zeroes of a polynomial are visual representations of the point on a graph where a curve intersects with the x-axis. In other words; If f(a)=0, a is a zero of polynomial f(x).

For example: Let’s take f(x) = x² - 4
To find the zero, set f(x) = 0
                                        x²- 4 = 0
                                                              (x - 2) (x + 2) = 0
So, the zeroes are x= 2 and x= -2

According to the theorem, polynomial f(x) has degree  1. This means that the highest power of variable x is 1. (x - a) is a factor of f(x) if f(a)=0. Here’ a’ is a real number.
So, the formula for factor theorem is:  f(x)=(x-a)q(x)

• (x-a) is a factor of f(x)
• f(a)=0
• q(x) represents the quotient polynomial, i.e., the result of dividing the original polynomial by the factor.
• When f(x) is divided by (x-a) the remainder is zero
• a is the solution to f(x) =0 and also the zero function of f(x).

As established above, the factor theorem is generally used while solving polynomial equations.

Let’s see how to apply it, using an example:

Question: Use the factor theorem to check whether (x-3) is a factor of 
                                       f(x) - x³ -6x² = 11x - 6
According to the theorem:
If f(3)=0, then, (x-3) is a factor of f(x)
         f(3) = (3)³ - 6(3)² + 11(3) - 6 
                  = 27- 54 + 33 - 6
                      = 0
Since f(3) = 0
(x-3) is a factor of the polynomial f(x) - x³ -6x²=11x-6
 

To factorize a cubic polynomial:
Find zero using the trail- and-error method. Then, using synthetic division method, divide the given polynomial f(x) by the given binomial (x-a), 

After division, if the remainder is not zero, then (x-a) is not a factor of f(x). 

If the remainder is zero, use the division algorithm and write the given polynomial as a product of (x-a) and quadratic quotient q(x); f(x) = (x-a)q(y) + r

If possible, factor the quadratic quotient further

Then, represent the polynomial in factored form.

Let’s factor f(x) = x³ - 6x² + 11x - 6 using the aforementioned procedure 

The first step is to find a zero using the hit and try method and dividing the given polynomial
Try x=1,
            f(1) = 13 - 6(1)² + 11(1) - 6 = 1 - 6 + 11 - 6 = 0    
So, x = 1 is a zero, a (x-1) is a factor.

Now, we will use synthetic division to divide f(x) by (x-1)
            
                  | 1   -6    11   -6
              1  |      1   -5     6   | 0
                  ---------------------
                    1   -5     6    0
 
The quotient is: x² - 5x = 6 

Since the remainder is zero, 

                 X² - 5x + 6 = (x-2) (x-3)
The polynomial, in its final factored form, is 
                                       f(x) = (x-1)(x-2)(x-3)

Thus, the zeroes of given polynomial x3 - 6x2 + 11x - 6 are 1, 2, and 3
 

The factor theorem is a mathematical concept helpful in practical fields like engineering, physics, computer graphics, economics, business marketing, robotics etc. Let’s discuss how:

• Analyzing modelling systems in engineering: Engineers use polynomials for analyzing and modelling systems. The factor theorem helps analyze stability or resonance points.

• Solving displacement points in physical systems: The factor theorem contributes in solving for critical values like time of flight, maximum height or zero- displacement points in physical systems like projectile motion, waveforms, or oscillations.

• Simplifying and computing transformations in computer graphics: Factorization helps simplify curve behavior for curves like Bézier curves, used to draw animations or 3D models; it also helps simplify curve behavior, find intersections and compute transformations efficiently. 

• Finding break-even points in economic and business modeling: Analysts use factor theorems to find break-even points and optimize production. Polynomial functions model profit, cost, and revenue functions.

• Breaking down complex polynomials in robotics and game development:The theorem breaks down complex polynomial-based decisions or movements into manageable parts. Robots and engines use these polynomials to calculate motion paths or AI decision models.