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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 ax3 + bx2 + 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.
|1.||Definition of Cubic Polynomial|
|2.||How to Solve Cubic Equation?|
|3.||Solving Cubic Equations Using Factoring|
|4.||Graph of Cubic Polynomial|
|5.||Roots of Cubic Polynomial|
|6.||FAQs on Cubic Polynomial|
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): ax3 + bx2 + 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): x3 − 5x2 + 15x − 6, r(z): πz3 + (√2)10.
Cubic Polynomial Formula
The cubic polynomial formula is in its general form: ax3 + bx2 + cx + d a cubic equation is of the form ax3 + bx2 + 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 ax3 + bx2 + 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 x3 - 2x2 - 8x - 35 = 0 if (x - 5) is a factor of the cubic polynomial x3 - 2x2 - 8x - 35.
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 x2 + 3x + 7 and the remainder is 0. Thus, by factor theorem, the given cubic equation can be written as: (x3 - 2x2 - 8x - 35) = (x - 5) (x2 + 3x + 7).
Now, we will solve x2 + 3x + 7 = 0 by the quadratic formula. Then we get:
x = [ (- 3 ± √ (32 - 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): a3 ± 3a2b + 3ab2 ± b3 = (a ± b)3
- Difference of the cubes: a3 − b3 = (a − b)(a2 + ab +b2)
- Sum of the cubes: a3 + b3 = (a + b)(a2 − ab + b2)
Example: Find the roots of cubic polynomial y3 – 2y2 – y + 2.
We will start by factorizing the equation:
y3 – 2y2 – y + 2 = y2(y – 2) – (y – 2)
= (y2 – 1) (y – 2)
= (y + 1) (y – 1) (y– 2) (∵ a2 - b2 = (a + b)(a - b))
y = 1, -1 and 2.
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 = ax3 + bx2 + 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 ax3 + bx2 + 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, x3 - 12x2 + 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, x3 - 12x2 + 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
3x = 12
x = 4.
We can find out the two roots by factorizing the equation into a quadratic equation. Look at the image below.
x2 - 8x + 7 = 0
x2 - 7x - x + 7 = 0
x(x - 7) - 1 (x - 7) = 0
(x - 1) (x - 7) = 0
x -1 = 0 and x - 7 = 0
x = 1 and x = 7
Therefore, the roots are 1, 4 and 7.
Important Notes on Cubic Polynomials:
- A cubic polynomial is of the form p(x) = ax3 + bx2 + 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.
Cubic Polynomial Examples
Example 1: Which of the following are cubic polynomials?
p(x): 5x2 + 6x + 1
q(z): z2 − 1
p(y): y3 − 6y2 + 11y − 6
q(y): 81y3 − 1
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): y3 − 6y2 + 11y − 6 and q(y): 81y3 − 1.
Answer: p(y) and q(y).
Example 2: Check whether 2y + 1 is a factor of the polynomial 4y3 + 4y2 – y – 1 or not using the factor theorem.
Let's equate the given binomial 2y + 1 = 0.
∴ y = -1/2
Substitute y = -1/2 in the given polynomial equation 4y3 + 4y2 – y – 1.
⟹ 4( -1/2)3 + 4(-1/2)2 – (-1/2) – 1
= -1/2 + 1 + 1/2 – 1
Answer: The remainder = 0, thus, 2y + 1 is a factor of the cubic polynomial 4y3 + 4y2 – y – 1.
Example 3: Find the roots of cubic polynomial f(x) = 3x3 - 5x2 + 4x + 2 by rational root theorem.
To find the roots of f(x), we have to solve the cubic equation 3x3 - 5x2 + 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,
3x2 - 6x + 6 = 0
Dividing both sides by 3,
x2 - 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.
Practice Questions on Cubic Polynomials
FAQs on Cubic Polynomials
What is Cubic Polynomial With Example?
A cubic polynomial is a type of polynomial with a degree of 3 i.e. the highest exponent of the variable is 3. The general form of a cubic polynomial is written as p(x): ax3 + bx2 + cx + d, a ≠ 0, where
- a, b, and c are coefficients and
- d is the constant with all of them being real numbers.
How Do You Solve Cubic Polynomials?
Cubic polynomials can be solved by converting a cubic equation into the product of a linear factor and quadratic equation. Solving a cubic polynomial is done by using factor theorem and synthetic division.
How to Use Factor Theorem for Cubic Equations?
To find the roots of a cubic equation, we first do trial and error by different linear factors to check whether they are factors of the cubic polynomial. To determine whether it is a factor, we use the factor theorem.
How Does Factorizing Cubics Help to Solve Cubic Equations?
Sometimes, factorizing cubics can be possible. For example, the cubic equation x3 + 3x2 - 4x - 12 = 0 can be solved using factorizing as follows: x2 (x + 3) - 4 (x + 3) = 0 ⇒ (x + 3) (x2 - 4) = 0 ⇒ (x + 3) (x - 2) (x + 2) = 0 ⇒ x = -3, 2, -2.
How to Use Synthetic Division Method for Cubic Polynomials?
Cubic polynomials are solved by the synthetic method using general steps like taking the coefficients alone, bringing the first down, multiplying with the zero of the linear factor, and adding with the next coefficient, and repeating until the end.
What is Cubic Root Formula?
The cubic formula is ax3 + bx2 + cx + d = 0. There is a wondering relation between the roots and the coefficients of a cubic polynomial. If α, β, and γ are the zeros of a cubic polynomial then the cubic root formulas are:
- α + β + γ = -b/a
- αβ + βγ + γα = c/a
- αβγ = -d/a
These are referred to as cubic equation roots formulas as they can be used to solve cubic equations at times.