The cubic equation formula is used to represent the cubic equation. A polynomial having degree three is known as a cubic polynomial or we can call it a cubic equation. Cubic equations have at least one real root and they can have up to 3 real roots. The roots of a cubic equation can be imaginary as well but at least 1 must be real. The cubic equation formula along with a few solved examples is explained below. Let us explore them.
What Is Cubic Equation Formula?
The cubic equation formula can also be used to derive the curve of a cubic equation. Representing a cubic equation using a cubic equation formula is very helpful in finding the roots of the cubic equation. A polynomial of degree n will have n number of zeros or roots. The cubic equation is of the following form:
ax3+bx2+cx+d=0
We can solve the cubic equation by two methods
i) Trial - Error and Synthetic Division
ii) Factorization.
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Let us see the applications of the cubic equation formula in the following solved examples.
Examples Using Cubic Equation Formula
Example 1: Select the cubic polynomials from the following:
- p(x): 5x2 + 6x + 1
- p(x): 2x + 3
- q(z): z2 − 1
- r(z): z2 + (√2)9
- r(z): √5z2
- s(x): 10x
- p(y): y3 − 6y2 + 11y − 6
- q(y): 81y3 − 1
- r(z): z + 3
Solution:The cubic polynomials among the above given polynomials are:
| Cubic Polynomials |
p(y): y3 − 6y2 + 11y − 6 q(y): 81y3 − 1 r(z): z2 + (√2)9 |
Example 2: Find the roots of the following cubic equation 2x3 + 3x2 – 11x – 6 = 0
Solution:
To find: Roots of the given equation.
This equation can not be solved using the factorization method, we will use the trial and error method to find one root.
We generally start with the value “1”.
f (1) = 2 + 3 – 11 – 6 ≠ 0
f (–1) = –2 + 3 + 11 – 6 ≠ 0
f (2) = 16 + 12 – 22 – 6 = 0
Value “2” makes the L.H.S equal to “0”. Hence two is one of the three roots.
Now we will use Synthetic Division Method to find the other two roots.
We divide our equation by (x-2) and the quotient will give us the other two roots. We divide our equation by (x-2) and the quotient will give us the other two roots.
Quotient : (2x2 + 7x + 3)
Factorising this quotient,
(2x+1) (x+3)
From here we get the values of x as,
x = -1/2 and x = -3
Answer: So, the three roots of the cubic equation are x = 2, x = -1/2 and x = -3
Example 3: Using the cubic equation formula, solve the cubic equation x3 – 2x2 – x + 2.
Solution:
To find: Roots of the above equation
We will first check whether we can factorize the cubic equation or not, if it can not be factorized we have to use the synthetic division method. But in this case, by inspection, we can tell that this equation can be solved by factorization. Let's see how.
x3 – 2x2 – x + 2.
= x2(x – 2) – (x – 2)
= (x2 – 1) (x – 2)
= (x + 1) (x – 1) (x – 2)
We can conclude that,
x = -1, x = 1 and x = 2.
Answer: So, the three roots of the cubic equation are x = -1, x = 1 and x = 2.
FAQs on Cubic Equation Formula
What Is Cubic Equation Formula?
The cubic equation formula can also be used to derive the curve of a cubic equation. Representing a cubic equation using a cubic equation formula is very helpful in finding the roots of the cubic equation. A polynomial of degree n will have n number of zeros or roots. The cubic equation is of the following form: ax3+bx2+cx+d=0
How to Solve Cubic Polynomials Using Cubic Polynomial Formula?
The most commonly used strategy for solving a cubic equation is
- Step 1: Reduce a cubic polynomial to a quadratic equation.
- Step 2: Solve the quadratic equation using the quadratic formula.
What Is the Equation for Cubic Polynomials Formula?
A cubic equation is an algebraic equation of degree three and is of the form ax3 + bx2 + cx + d = 0, where a, b and c are the coefficients and d is the constant.
