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 \(ax^3 + bx^2 + cx + d\). Here, we can see that the highest power of the variable x is 3. The coefficients, a, b, and c, and the constant d, are real numbers and \(a ≠ 0\). A cubic equation is an equation that contains a cubic polynomial.

The general formula that we use for a cubic polynomial is \(f(x) = ax^3 + bx^2 + cx + d\), 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 \(ax^3 + bx^2 + cx + d = 0\), 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 \(ax^3 + bx^2 + cx + d = 0\). If the equation is not in standard form, for example, \(bx^2 + ax^3 + cx + d = 0\), then we should rearrange 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, by solving \(x^3 + 2x^2 - 5x - 6 = 0.\)

Step 1: As the given polynomial is in standard form 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 \(x^3 + 2x^2 - 5x - 6 = 0\), we get:

\(1^3 + 2(1)^2 - 5(1) - 6\) 

\(= 1 + 2 - 5 - 6     \)
\(= -8\)

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:

\((x^3 + 2x^2 - 5x - 6) \div (x + 1) \)

By using the long division method:

\(x^3 \div x = x^2\)

Multiplying the answer by the denominator, we get:

\(x^2(x + 1) = x^3 + x^2\)

Now, subtract \(x^3 + x^2 \) from \(x^3 + 2x^2\)

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

The next step is to bring down -5x and repeat the process

Now, dividing the second term:

\(x^2 ÷ x = x\)

Now multiply x by \(x + 1\)

\(x(x + 1) = x^2 + x\)

Subtract \(x^2 + x\) from \(x^2 - 5x\) 

\((x^2 - 5x) - (x^2 + x) = -6x \)
Bring down -6 and repeat the process

Divide the third term

\(-6x \div x = -6 \)
Multiply -6 by x + 1
\(-6(x + 1) = -6x - 6\)

Subtracting, we get:

\((-6x - 6) - (-6x - 6) = 0\)

Therefore, \({{x^3 + 2x^2 - 5x - 6} \over {x + 1 } } = {x^2 + x -6}\)

Step 3: The last step is to factor the quadratic. 

So, \(x^2 + x - 6 = (x + 3) (x - 2)\)

Therefore, the factors are \((x + 1) (x + 3) (x - 2)\). 

So x = -1, -3, 2.
 

Synthetic division is a quick and effective method to reduce a cubic polynomial to a quadratic. The quadratic can then be solved easily. Here, the division of two polynomials can be represented in the form \({p(x) \over q(x)} = Q + {R \over q(x)}\).
Here, p(x) is the dividend
q(x) is the linear divisor.
Q is the quotient
R is the remainder

The steps involved in the synthetic division method are:  

Step 1: Make sure the polynomial is in the standard form. If not re-arrange them in standard form, even if the coefficients are zero.

For example, \(x^3 + 2x - 4 \)should be rewritten as \(x^3 + 0x^2 + 2x - 4\). 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: Write the coefficient of the polynomial in a row. Write r (the zero) to the left. 

Step 4: Bring the first coefficient straight down as it is 

Step 5: Now multiply the number just brought down by r, and write the result under the next coefficient and add. Repeat the process for all coefficients. 

Step 6: Group the coefficients and the variables together to get the quotient. The last number is the remainder and other numbers gives the quotient, which is one degree less than the original polynomial. 

Step 7: Use factorization or quadratic formula to find the remaining roots. 
 

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 \(p(x) = ax^3 + bx^2 + cx + d\):

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 > 0, the graph starts from the third quadrant and rises to the first quadrant.
• If a < 0, the graph starts from the top-left and falls to the bottom-right.

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, with one root having 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 cubic polynomials helps students understand complex curves, solve real-world problems, and develop analytical skills. Here are a few tips and tricks:

Before solving a cubic polynomial, always check if there is a common factor among all terms. Factorization simplifies the cubic equation and often reduces it to a quadratic, which makes it easier to solve. 

• To solve the cubic polynomials, students can use a graph; the point where the graph crosses the x-axis is the solution.
   
• To solve the complex cubic polynomial, use the synthetic division method. Use this method to reduce the cubic to a quadratic and then solve the polynomial. 
   
• Practice regularly makes it easier for students to understand the concept and helps them to solve even complex polynomials. 
   
• When solving cubic polynomials, always notice patterns. For example, \({x^3 + 3x^2 + 3x + 1} = {(x + 1)^3}\). Recognizing the perfect cubes helps to solve the polynomials faster. 

Cubic polynomials are used to model and solve many real-world problems involving curved or nonlinear relationships. Understanding how to use cubic polynomials helps in several fields beyond academics.

• In engineering and physics, cubic polynomials are used to analyze how structures bend or deform under load. For example, beam deflection formulas often involve cubic terms to describe how a beam curves when weight is applied. 
   
• Cubic polynomials are used in computer graphics, especially in Bezier curves and spline modeling, to create smooth and natural-looking curves. These equations are used in drawing smooth transitions in animations, 3D modeling, and video game design. 
   
• In economics, cubic polynomials are used to model cost, revenue, and profit curves when growth or decline rates are not constant. For example, a cubic polynomial can represent a company's profit when it increases, then decreases, and then rises again due to market changes. 
   
• Cubic polynomials are used in automotive design to model acceleration and braking curves of vehicles. Engineers use them to ensure smoother speed transitions and better fuel efficiency during acceleration or deceleration. 
   
• Architects use cubic equations to design arches, domes, and bridges that need specific curvature for strength and balance.