Factoring Cubic Polynomials

Cubic polynomials are such polynomials that have a degree of three, expressed by ax³ + bx² + cx + d (standard form of polynomial). For example, 3x³ + 3x² + 4x + 3. When the constant term(d) in cubic polynomials equals to zero, such polynomials are known as cubic trinomial.

Definition of Cubic Trinomial: A polynomial that has exactly three terms with a degree of three is known as cubic trinomial. ‘Cubic’ refers to the highest degree of three, and the word ‘trinomial’ suggests the expression has three terms.

The standard form of a cubic trinomial is:
ax³ + bx² + cx,

where,
 

• the degree of the polynomial is three because the highest power of the polynomial is three.
• The coefficients a, b, and c can be any real or complex numbers.
• x is the variable.

3x³ + 2x² + 4x is an example of a cubic trinomial.

Factorizing a cubic trinomial means breaking down a cubic trinomial into its factors. This process is important for solving cubic equations. The following steps are used to break down a polynomial of the form ax³ + bx² + cx + d.

• Step 1: Check for common factors.
  
  Check for the common factors in the given polynomial. If there is a common factor, write it down so that we will get a quadratic equation, which will help us to solve the equation easily.
  
  Example: \(x^3 + 4x^2 + 3x\)
  
  In the given polynomial, we have the common term x. So we have to factor out x first.
  \(x^3 + 4x^2 + 3x = x (x^2 + 4x + 3)\).
  
  Now we have a quadratic equation as: \(x^2 + 4x + 3\).

• Step 2: Factor the quadratic.
  
  After taking out the common factor, we are left with a quadratic equation. Now, we have to factorize that quadratic equation.
  
  \(x^2 + 4x + 3\)
  
  In this equation, we have to look at two numbers; the sum of the numbers is 4, and the product of the numbers is 3.
  
  Therefore, we get an equation as, \(x^2 + 4x + 3 = (x + 1)(x + 3)\).

• Step 3: Write the complete factorization.
  
  Now we have to put the factorization of the quadratic equation to step 1,
  \(x(x^2 + 4x + 3 )= x(x + 1)(x + 3)\).

• Step 4: Verify the factorization.
  
  Multiply the factors; if this matches the original polynomial expression, then the factorization is correct. 
   

1. First, multiply the factorization of the quadratic polynomial, we get
   \( (x + 1)(x + 3) = x² + 3x + x + 3\\   = x^2 + 4x + 3\)
    
2. Then multiply it by the x term, which we take as common in the first step, we get
   \(x(x^2 + 4x + 3) = x^3+ 4x^2 + 3x\)

This matches the original polynomial, so the factorization is correct.

Parent Tips: To get with the idea of factoring trinomials before moving to factoring cubic trinomials, children can use a factoring calculator to practice factorization of quadratic polynomials.

According to the rational root theorem, the possible roots of a cubic polynomial \(f(x) = ax^3 + bx^2 + cx + d\) can be determined by:

\(\text{Possible rational roots} = ±\frac{\text{factors of d}}{\text{factors of a}}\)

The following steps will help us understand the rational root theorem better:

• Step 1: Find the possible roots.
  
  Use the rational root theorem to find possible rational roots using the constant term d and leading coefficient a.
  
  For example,\( x^3 - 2x^2 - 5x + 6\).
  
  To find the possible roots, we have to divide the factors of d (constant term) by the factors of a (leading coefficient).
  
  Here,
   

1. the coefficient d = 6
2. Leading term a = 1
3. Factors of 6: ±1, ±2, ±3, ±6
4. Factors of 1: ±1
   
   So, the possible roots are: \(±\frac{\text{factors of d}}{\text{factors of a}} = \) ±1, ±2, ±3, ±6.

• Step 2: Test the possible roots.
  
  Now we have to check all the possible roots of the polynomial. If the value is 0, that is considered as a root. 
   

1. Try x = 1 in the polynomial
   
   \(f(x) = x^3 - 2x^2 - 5x + 6\\ f(1) = (1)^3 - 2(1)^2 - 5(1) + 6\\         = 1 - 2 - 5 + 6 = 0\)
   
   So, x = 1 is a root, and (x - 1) is a factor.

• Step 3: Use polynomial division.
  
  We divide the polynomial by the factor that we got by using the long division or the synthetic division method. 
  
  \((x^3 - 2x^2 - 5x + 6) ÷ (x - 1) \)
  
  After dividing this, the equation will become: \(x² - x - 6\).

• Step 4: Factor the quadratic equation.
  
  We got a quadratic expression after dividing the polynomial by the factor. Now we have to factorize that quadratic equation.
  
  \(x² - x - 6\) is the quadratic equation.
   

1. Find two numbers whose sum is -1 and product is -6.
   
   The numbers are -3 and 2
   \(x^2 - x - 6 = (x - 3)(x + 2).\)

• Step 5: Write the final factorization.
  
  After factorizing the equation, we have to combine all the factors to get the final answer. 
  
  \(f(x) = (x - 1)(x - 3)(x + 2)\\ x^3 - 2x^2 - 5x + 6 = (x - 1)(x - 3)(x + 2)\)

We can use special math rules called identities when a cubic polynomial has only two terms. Mentioned below are two main cases:

• Case 1: No constant terms.
  
  If there are no constant terms and the polynomial equation consists of only variable terms, the expression might look like,
  
  \(ax^3 + bx^2 \ or\ ax^3 + cx\)
  
  In this case, we can take the common x terms,
  
  \(ax^2  = x^2 (ax + b)\\ ax^3 + cx = x(ax^2 + c)\)

• Case 2: Equations that include constants 
   

1. If one term is just a number like ax³ + d, then we will use a special identity to factor:
    
2. If both the terms are perfect cubes, like 8x³ + 27, use:
   \(a^3 + b^3 = (a + b)(a^2 - ab + b^2)\)
    
3. If both the terms are cubes but with a minus, like x³ - 8, use:
   \(a^3 - b^3 = (a - b)(a^2 + ab + b^2)\)
   
   These help us to break the expression into smaller parts. 

Children from small grades might find factorization difficult. To make this easy for them, here a few simple tips and tricks:

1. Children can use factor calculator for finding factors of any number.
    
2. To check if you have the correct answer when factoring quadratic equations, children can use a quadratic factoring calculator to verify answers.
    
3. Don't always depend upon an online calculator, first manually solve the equation yourself, then use the calculator to verify the answers or if you are stuck.
    
4. For dividing polynomials, use long division method.
    
5. Use correct formula for finding cubes. Practice daily to memorize formulas.

Factoring cubic polynomials is used in many real-life applications, especially in fields like engineering, economics, physics, and computer science. Some of the applications are mentioned below:

1. Designing Roller Coaster: Cubic equations help shape the curves of roller coasters. Factoring is used to decide where the roller coaster should have ascents, descents, and changes of direction to make the ride enjoyable. 
   
    
2. Planning Car Speed: When pressing the pedal, a car's acceleration can be modeled by a cubic equation. Factoring helps determine the speed at different times.
   
    
3. Making Video games: designers use curves to make the character move smoothly in video games. These curves use cubic equations. 
   
    
4. Finding Volume: 3D shapes like cubes, cuboid, cones, tanks, etc. have volume that are expressed using cubic polynomials. By factoring these equations, we find the possible dimensions of the shapes.
   
    
5. Economics: In business, profits, cost and revenues models are represented with cubic polynomials. Factorization provides optimum production levels and maximizes profits, reducing losses.