Cubic Polynomial

A polynomial is a mathematical expression that consists of constants, variables, and exponents, depending on its type. For example, a type of polynomial where the highest degree of the variable is 3 is called a cubic polynomial. It is expressed as ax3 + bx2 + cx + d; we can see here that the highest power of variable x is 3. The coefficients, a, b, and c, and the constant d, are real numbers and a  0. A cubic equation involves a cubic polynomial, which is often expressed as a formula.
 

The general formula that we use for a cubic polynomial is f(x) = ax3 + bx2 + cx + d, where the cubic equation f(x) = 0. If the values of x manage to satisfy the equation where f(x) becomes 0, then they are called the roots. In the next section, we will learn how to solve a cubic equation.
 

If we want to solve a cubic equation of the form ax3 + bx2 + cx + d = 0, we need to find the values of x to make the equation true. There are many methods to solve cubic equations, but today we’ll be focusing on the factorization method. The steps involved in this method are given below:

Step 1: Rearrange the equation 
The equation must be written in the standard form, which is ax3 + bx2 + cx + d = 0. If the equation is bx2 + ax3 + cx + d = 0, then we should rearrange and ensure the equation is in the standard form.

Step 2: Use trial and error to find a rational root
Here, we should guess the value of one rational root. The value could be x = 1, -1, 2, -2, and so on. After finding one rational root, we should use polynomial division to break the equation into:
(x − r) (quadratic equation). Alternatively, we can also use the rational root theorem to find all possible rational roots.

Step 3: Factor the quadratic
We can solve the equation to factor the quadratic or use the formula. 

Let’s now apply these steps to solve a cubic equation. Let’s solve
x3 + 2x2 - 5x - 6 = 0.

Step 1: It is already in standard form, so we don’t have to rearrange.

Step 2: We will use trial and error to find a root. Let’s try x = 1. So substituting the value of x as 1 in the equation x3 + 2x2 - 5x - 6 = 0, we get:
   13 + 2(1)2 - 5(1) - 6 
= 1 + 2 - 5 - 6    
= -8
So, x = 1 is not a root. Let us now try x = -1. Substituting the value, we get:
  (-1)3 + 2(-1)2 - 5(-1) - 6
= -1 + 2 + 5 - 6
= 1 + 5 - 6
= 6 - 6 
= 0. 
If x = -1 is a root, then (x + 1) is a factor. Now, we should divide the polynomial by (x + 1) to break the equation. 
To do so, let’s set up the division:
(x3 + 2x2 - 5x - 6)  (x + 1) 
By using the long division method:
x3  x = x2
Multiplying the answer by the denominator, we get:
x2(x + 1) = x3 + x2
Now, subtract x3 + x2 from x3 + 2x2
(x3 + 2x2) - (x3 + x2) = x2
The next step is to bring down -5x and repeat the process
Now, dividing the second term:
x2  x = x
Now multiply x by x + 1
x(x + 1) = x2 + x
Subtract x2 + x from x2 - 5x 
(x2 - 5x) - (x2 + x) = -6x
Bring down -6 and repeat the process
Divide the third term
-6x  x = -6
Multiply -6 by x + 1
-6(x + 1) = -6x - 6
Subtracting, we get:
(-6x - 6) - (-6x - 6) = 0
Therefore, x3 + 2x2 - 5x - 6x + 1 = x2 + x - 6

Step 3: The last step is to factor the quadratic. 
So, x2 + x - 6 = (x + 3) (x - 2)
Therefore, the factors are (x + 1) (x + 3) (x - 2). 
So x = -1, -3, 2.
 

In this method, we use linear binomials of the form x - r to divide polynomials. Here, r is a root. Synthetic division is a quick and effective way to reduce a cubic polynomial into a quadratic, which can be solved easily. Here, the division of two polynomials can be represented in the form p(x) / q(x) = Q + R / q(x). Here, p(x) is the dividend and q(x) is the linear divisor. As we all know, Q and R stand for quotient and remainder, respectively. The steps involved in the synthetic division method are:  

Step 1: Make sure the polynomial is in the standard form. Re-arrange if required and ensure all terms are present, even if the coefficients are zero.
For example, x3 + 2x - 4 should be rewritten as  x3 + 0x2 + 2x - 4. Here, 1, 0, 2, -4 are the coefficients. 

Step 2: We will now perform the division by considering the coefficients as the dividend and zero of the linear factor (x - r) as the divisor. Naturally, the coefficients will take the dividend’s place. Here, the divisor is a linear factor: q(x) = x - r and r is the value that makes the divisor zero. Now write r in the divisor’s place.

Step 3: Bring down the first coefficient and then multiply it by the zero of the linear factor. Write down the result below the next coefficient.

Step 4: Add them and record the answer. We should now repeat the previous 2 steps and continue until we reach the last term. 

Step 5: By now, we should have a row of numbers. We need to identify and isolate the last number, which will be our remainder. 

Step 6: Group the coefficients and the variables together to get the quotient.
 

The factor theorem helps identify factors and roots, which simplifies the factoring of the polynomial. Let’s see how the factor theorem can be used to solve a cubic polynomial p(x) = ax³ + bx² + cx + d:

Step 1: Identify possible rational roots by using the rational root theorem. If required, use the synthetic division method to see if the remainder is 0. 

Step 2: Confirm that x - r is a factor by checking if the remainder from synthetic division is 0. 

Step 3: Divide p(x) by x - r by using the synthetic division method to get a quadratic quotient. Once the quadratic quotient is found, the given polynomial can be written as p(x) = (x - r) (quadratic). 

Step 4: Solve the quadratic further if the remainder is not equal to zero and find the remaining roots.
 

A cubic polynomial can also be solved with the help of graphs. The roots of the polynomial can be found by looking at the points where the graph cuts or crosses the x-intercept. There are two things we should remember while plotting the graph:

If a is positive, then the graph begins in the third quadrant and ends in the first quadrant.
If a is negative, then the graph will be drawn from up to down.
Here is a step-by-step process on how to sketch the graph of a cubic polynomial:

Step 1: The first step is to find the y-intercept. In other words, we should evaluate p(0).

Step 2: Use the rational root theorem to find the roots. Once the roots are found, we can apply synthetic division to confirm the roots. Once we confirm the roots, we can factor the polynomial.

Step 3: Check the sign of a, which will later decide how the graph looks.

Step 4: Plot the points obtained from the calculations. After this, we can draw a smooth curve to finish the graph.

A cubic polynomial’s roots represent the values of x in such a way that they equate the polynomial to zero. Usually, a cubic polynomial will have three roots, including complex roots. Their properties are as follows:

• All three roots can be distinct and real. For example, x = 1, 2, 3.
• If a polynomial has real coefficients, then the complex roots will always appear in conjugate pairs. For example, if a + bi is a root, then a - bi is also a root.
• Two roots can be complex and one can be real. 
• Two real roots, but one root has multiplicity 2.
• One real root with multiplicity 3.

If the multiplicity is 1, then the graph cuts the x-axis. For multiplicity 2, the graph touches the x-axis and turns back. When multiplicity is 3, the graph flattens at the x-axis. It then crosses the x-axis, giving rise to an S-shaped curve.
 

Learning to solve for cubic polynomials has more advantages than just scoring good grades in exams. It has several real-life applications, and we’ll be looking at them one by one:

• Engineering: Cubic equations are used in engineering during stress/strain analysis. These equations are used in materials science to check the different behavioral properties of materials under different forces.
  
   
• Economics: Cubic polynomials are used in the field of economics to describe complex cost and profit functions with increasing or decreasing rates. 
  
   
• Automobile: cubic functions are used to model acceleration curves in cars. This is done to make sure the car has better acceleration.
  
   
• Architecture: Cubic deflection equations are often used to bend the beams under loads while designing bridges or similar structures.    
  
       
• Music: Sound engineers use cubic polynomials to work with sound waves. They can also use them to effectively modulate audio effects.