Cubic Polynomial

A cubic polynomial is a type of polynomial with the highest power of the variable or degree to be 3. It is of the form ax 3 + bx 2 + cx + d. Here, 'x' is a variable and a, b, c, and d are real numbers. Cubic polynomials are used in various areas of mathematics and science, including physics, engineering, and economics.

A polynomial is an algebraic expression with variables and constants with exponents as whole numbers. Let us learn more about cubic polynomials, the definition, the formulas, and solve a few examples.

Definition of Cubic Polynomial

A cubic polynomial is a polynomial with the highest exponent of a variable i.e. degree of a variable as 3. Based on the degree, a polynomial is divided into 4 types namely, zero polynomial, linear polynomial, quadratic polynomial, and cubic polynomial. The general form of a cubic polynomial is p(x): ax 3 + bx 2 + cx + d, a ≠ 0, where a, b, and c are coefficients and d is the constant with all of them being real numbers. An equation involving a cubic polynomial is called a cubic equation. Some of the examples of a cubic polynomial are p(x): x 3 − 5x 2 + 15x − 6, r(z): πz 3 + (√2) 10 .

Cubic Polynomial Formula

The cubic polynomial formula is in its general form: ax 3 + bx 2 + cx + d a cubic equation is of the form ax 3 + bx 2 + cx + d = 0. The values of 'x' that satisfy the cubic equation are known as the roots/zeros of the cubic polynomial. Let us see how to find them in different ways.

How to Solve Cubic Equation?

Solving a cubic polynomial is nothing but finding its zeros. The general form of a cubic equation is ax 3 + bx 2 + cx + d = 0, a ≠ 0. To solve a cubic equation:

• Step 1: Re-arrange the equation to standard form
• Step 2: Break it down to the product of linear factor and quadratic equation
• Step 3: Then solve the quadratic equation

Here, Step 2 can be done by using a combination of the synthetic division method and the factor theorem. Let us see how to solve cubic equations using these steps.

☛Note: Alternatively, a cubic equation can be solved by the rational root theorem. To understand it fully, click here.

Cubic Polynomial and Synthetic Division

Synthetic division is a method used to perform the division operation on polynomials when the divisor is a linear factor. We can represent the division of two polynomials in the form: p(x)/q(x) = Q + R/(q(x))

• p(x) is the dividend
• q(x) is the linear divisor
• Q is quotient
• R is remainder

While solving a cubic polynomial we use the synthetic division method and the steps are:

• Step 1: Check whether the cubic polynomial is in the standard form.
• Step 2: Write the coefficients in the dividend's place and write the zero of the linear factor in the divisor's place.
• Step 3: Bring the first coefficient down, multiply it with zero of linear factor and write it below the next coefficient.
• Step 4: Add them and write the value below.
• Step 5: Repeat the previous 2 steps until you reach the last term.
• Step 6: Separate the last term thus obtained which is the remainder.
• Step 7: Now group the coefficients with the variables to get the quotient.

Cubic Polynomial and Factor Theorem

Factor theorem is a that links the factors of a polynomial and its zeros. As per the factor theorem, (x – a) can be considered as a factor of the polynomial p(x) of degree n ≥ 1, if and only if p(a) = 0. Here, a is any real number. The formula of the factor theorem is p(x) = (x – a) q(x). It is important to note that all the following statements apply for any polynomial p(x) when (x – a) is a factor of p(x).

• p(a) = 0.
• The remainder is zero when p(x) is divided by (x – a).
• The solution to p(x) = 0 is "a" and the zero of the function p(x) is "a".

For degrees like 3 and 4, such as a cubic equation, factor theorem is used along with synthetic division and the steps are as follows:

• Step 1: Use the synthetic division method to divide the given polynomial p(x) by the given binomial (x − a)
• Step 2: Once the division is completed the remainder should be 0. If the remainder is not zero, then it means that (x - a) is not a factor of p(x).
• Step 3: Using the division algorithm, write the given polynomial as the product of (x - a) and the quadratic quotient q(x)
• Step 4: If it is possible, factor the quadratic quotient further.
• Step 5: Express the given polynomial as the product of its factors.

Example: Solve cubic equation x 3 - 2x 2 - 8x - 35 = 0 if (x - 5) is a factor of the cubic polynomial x 3 - 2x 2 - 8x - 35.

Solution:

The polynomial is of order 3. The divisor is a linear factor. Let's use synthetic division to find the quotient.

Here, the zero of the linear factor is found by: x - 5 = 0 ⇒ x = 5.

Thus, the quotient is one order less than the given polynomial. It is x 2 + 3x + 7 and the remainder is 0. Thus, by factor theorem, the given cubic equation can be written as: (x 3 - 2x 2 - 8x - 35) = (x - 5) (x 2 + 3x + 7).

Now, we will solve x 2 + 3x + 7 = 0 by the quadratic formula. Then we get:

x = [ (- 3 ± √ (3 2 - 4 (1)(7)) ] / (2(1))
= [ -3 ± √-19 ] / 2
= (-3 ± i √19) /2

Thus, the roots of given cubic equation are: 5, (-3 + i √19) /, and (-3 - i √19) /2.

Solving Cubic Equations Using Factoring

Cubic polynomials can be solved using factorization as well. But to make it to a much simpler form, we can use some of these special products:

• Perfect cube (2 forms): a 3 ± 3a 2 b + 3ab 2 ± b 3 = (a ± b) 3
• Difference of the cubes: a 3 − b 3 = (a − b)(a 2 + ab +b 2 )
• Sum of the cubes: a 3 + b 3 = (a + b)(a 2 − ab + b 2 )

Example: Find the roots of cubic polynomial y 3 – 2y 2 – y + 2.

We will start by factorizing the equation:

y 3 – 2y 2 – y + 2 = y 2 (y – 2) – (y – 2)

= (y + 1) (y – 1) (y– 2) (∵ a 2 - b 2 = (a + b)(a - b))

Graph of Cubic Polynomial

A cubic polynomial function of the third degree has the form shown on the right and it can be represented as y = ax 3 + bx 2 + cx + d, where a, b, c, and d are real numbers and a ≠ 0. When a cubic polynomial cannot be solved with the above-mentioned methods, we can solve it graphically. The points where the graph crosses the x-axis (x-intercepts) are considered the solution and are called the roots of a cubic polynomial. While plotting a graph for a cubic polynomial, we need to remember two important aspects:

• If the sign of "a" is positive, then the graph will be from down to up.
• If the sign of "a" is negative, then the graph will be up to down.

The graph of a cubic polynomial looks like this:

Roots of Cubic Polynomials

The solutions to a cubic equation are called the roots of the cubic equation. In most cases, there are 3 roots of a cubic polynomial but sometimes we do get two or only one. When a cubic polynomial is solved graphically, we get to the accurate roots or when we solve the equation with the formula, we derive the roots. Let us say p,q, and r are 3 roots for the equation ax 3 + bx 2 + cx + d. The formulas that indicate the relation between roots and the coefficients of a cubic polynomial are:

• p + q + r = - b/a
• pq + qr + rp = c/a
• pqr = - d/a

Example: Solve the cubic equation, x 3 - 12x 2 + 39x - 28 = 0 given that its roots are in arithmetic progression.

Solution: Let us consider 3 roots in AP to be x - d, x, and x + d.

p = x - d, q = x, r = x + d

From the equation, x 3 - 12x 2 + 39x - 28 = 0 we know,

a = 1, b = - 12, c = 39, d = - 28

By above cubic formulas:

Sum of the roots = - b/a

x - d + x + x + d = - (-12)/1

We can find out the two roots by factorizing the equation into a quadratic equation. Look at the image below.

x 2 - 7x - x + 7 = 0

x(x - 7) - 1 (x - 7) = 0

x -1 = 0 and x - 7 = 0

Therefore, the roots are 1, 4 and 7.

Important Notes on Cubic Polynomials:

• A cubic polynomial is of the form p(x) = ax 3 + bx 2 + cx + d
• The values of 'x' that satisfy the equation p(x) = 0 are called roots of cubic polynomial p(x).
• Since the degree of p(x), the cubic equation p(x) = 0 can have a maximum of 3 roots.
• The roots of a cubic equation can be found by using synthetic division, factoring, or rational root theorem.

☛Related Topics:

• Roots of Cubic Equation Calculator
• Solve Cubic Equation Calculator
• Polynomial Function
• Linear, Quadratic, and Cubic Polynomials

Cubic Polynomial Examples

Example 1: Which of the following are cubic polynomials? p(x): 5x 2 + 6x + 1 q(z): z 2 − 1 p(y): y 3 − 6y 2 + 11y − 6 q(y): 81y 3 − 1 Solution: A polynomial is said to be cubic only if its degree is 3. From the four polynomials, only two are cubic polynomials. They are: p(y): y 3 − 6y 2 + 11y − 6 and q(y): 81y 3 − 1. Answer: p(y) and q(y).

Example 2: Check whether 2y + 1 is a factor of the polynomial 4y 3 + 4y 2 – y – 1 or not using the factor theorem. Solution: Let's equate the given binomial 2y + 1 = 0. ∴ y = -1/2 Substitute y = -1/2 in the given polynomial equation 4y 3 + 4y 2 – y – 1. ⟹ 4( -1/2) 3 + 4(-1/2) 2 – (-1/2) – 1 = -1/2 + 1 + 1/2 – 1 = 0 Answer: The remainder = 0, thus, 2y + 1 is a factor of the cubic polynomial 4y 3 + 4y 2 – y – 1.

Example 3: Find the roots of cubic polynomial f(x) = 3x 3 - 5x 2 + 4x + 2 by rational root theorem. Solution: To find the roots of f(x), we have to solve the cubic equation 3x 3 - 5x 2 + 4x + 2 = 0. By the rational root theorem, the possible rational roots of f(x) are ± 1, ±2, ± 1/3, ± 2/3. By finding the value of f(x) at each of these values, we can easily see that f(-1/3) = 0 because f(-1/3) = 3(-1/3) 3 - 5(-1/3) 2 + 4(-1/3) + 2 = 0 Now, using synthetic division: Using the remainder, 3x 2 - 6x + 6 = 0 Dividing both sides by 3, x 2 - 2x + 2 = 0 By quadratic formula, x = (2 ± √(-4)) / 2 = (2 ± 2i) /2 = 1 ± i Answer: The zeros of the given cubic polynomial are -1/3, 1 - i, and 1 + i.

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