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 find remainders without performing long division. It states that when a linear divisor (x-a) divides a polynomial f(x), the remainder is f(a). 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 \(f(x) = x^3 -6x^2 + 11x - 6\).

  Check \(f(1)\),
 \(f(1) = 1^3 -  6(1)^2 + 11(1) - 6 = 0\)
 

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

Proof of Factor Theorem: 

According to division algorithm for polynomials:

\(f(x) = {(x - a)} \cdot {q(x)} + r\)

Where:

\(f(x)\) is the original polynomial
\((x - a)\) is the divisor
\(q(x)\) is the quotient polynomial
r is the remainder   

Substituting \(x = a\)
\(f(a)= (a - a)  \cdot q(a) + r =  r    \)       
Thus, substituting \(x = a\), we get \(f(a) = r\)
If \(f(a) = 0\), then \(r = 0\)
\(f(x) = (x-a)  \cdot  q(x)\)

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. The zeroes of a polynomial correspond to points where its graph intersects the x-axis. In other words; If f(a)=0, a is a zero of polynomial f(x).

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

According to the theorem, for any polynomial \(f(x)\), if \(f(a) = 0\), then \((x - a)\) is a factor \((x - a)\) 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 
                                   \(f(x) - x^3 -6x^2 = 11x - 6\)

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 \(f(x) - x^3 -6x^2 \\ =11x-6\)

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 \(f(x) = x^3 - 6x^2 + 11x - 6\) using the aforementioned procedure 

The first step is to find a zero using the hit and try method and dividing the given polynomial
Try \(x = 1\),
            \(f(1) = 1^3 - 6(1)^2 + 11(1) - 6\\ = 1 - 6 + 11 - 6\\ = 0 \)

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)
            \(\begin{array}{r|rrrr} & 1 & -6 & 11 & -6 \\ 1\, & & 1 & -5 & 6 \\ \hline & 1 & -5 & 6 & 0 \\ \end{array} \)
 
The quotient is: \(x^2 - 5x = 6 \)

Since the remainder is zero, 

                 \(x^2 - 5x + 6 = (x-2) (x-3)\)

The polynomial, in its final factored form, is 

                    \(f(x) = (x-1)(x-2)(x-3)\)

Thus, the zeroes of the given polynomial \(x^3 - 6x^2 + 11x - 6 \)are 1, 2, and 3
 

To help students confidently apply the Factor Theorem, it’s important to understand both the concept and the process. These tips and tricks will help students in identifying factors, verifying zeros, and avoiding common mistakes while solving polynomial problems efficiently.

• Understand the connection between zeros and factors. So remember that if \(f(a) = 0,\) then \(x - a\) is a factor of \(f(x)\).
  
   
• When checking for possible zeros, always start checking with small integers like \(\pm 1,\ \pm 2,\ \pm 3,\ \pm 6\). To make the calculation easier. 
  
   
• After finding one factor, use synthetic division to simplify the polynomial and find other factors. 
  
   
• Always double-check when substituting and calculating to avoid errors. 
  
   
• Visually represent the polynomial as a graph to avoid the confusion and to find the zeros. 

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:

• Engineering System Analysis: Engineers use polynomials to model and analyze systems. The Factor Theorem helps determine stability and identify critical points such as resonance or equilibrium.

• Physical Systems and Displacement: 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.

• Computer Graphics and Animation: In graphics, factorization simplifies curves, such as Bézier curves, used in animations and 3D modeling. It helps find intersections, predict curve behavior, and perform transformations efficiently.

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

• 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.