Solving Cubic Equation Using Factors
The solution of cubic equation using factor theorem is explained using the example added below,
Example: Find the roots of equation f(x) = 3x3 −16x2 + 23x − 6 = 0.
Solution:
Given expression: f(x) = 3x3 −16x2 + 23x − 6 = 0
First, factorize the polynomial to get roots
Since the constant is -6 the possible factors are 1, 2, 3, 6
f(1) = 3 – 16 + 23 – 6 ≠ 0
f(2) = 24 – 64 + 46 – 6 = 0
f(3) = 81 – 144 + 69 – 6 = 0
f(6) = 648 – 576 + 138 – 6 ≠ 0
We know that, iaccording to Factor Theorem if f(a) = 0, then (x-a) is a factor of f(x)
So, (x – 2) and (x – 3) are factors of f(x). Therefore, the product of (x – 2) and (x – 3) will also be factor of f(x). Now to find the remaining factors use the long division method and divide f(x) by product of (x – 2) and (x – 3)
Hence, Divisor = (x – 2)(x – 3) = (x2 – 5x + 6) and Dividend = 3x3 −16x2 + 23x − 6. Now divide as shown below,
After division we obtain (3x- 1) as quotient and remainder is 0. Now as per Division Algorithm we know that Dividend = Divisor×Quotient+Remainder.
⇒ f(x) = (3x3 −16x2 + 23x − 6) = (x2 – 5x + 6)(3x-1)
Since f(x) = 0
⇒ (x2 – 5x + 6)(3x-1) = 0
⇒ x2 – 5x + 6 = 0 or 3x-1 = 0
Now we will take 3x-1 = 0 ⇒ x = 1/3 as we already know two roots from x2 – 5x + 6 which are 2 and 3
So,
Roots of the given Cubic Equation are 1/3, 2, and 3.
Solving Cubic Equations
Cubic Equation is a mathematical equation in which a polynomial of degree 3 is equated to a constant or another polynomial of maximum degree 2. The standard representation of the cubic equation is ax3+bx2+cx+d = 0 where a, b, c, and d are real numbers. Some examples of cubic equation are x3 – 4x2 + 15x – 9 = 0, 2x3 – 4x2 = 0 etc.
Table of Content
- Polynomial Definition
- Degree of Equation
- Cubic Equation Definition
- How to Solve Cubic Equations?
- Solving Cubic Equations
- Solving Cubic Equation Using Factors
- Solving Cubic Equation Using Graphical Method
- Problems Based on Solving Cubic Equations
- Practice Problems on Solving Cubic Equations
For learning How to Solve Cubic Equations we must first learn about polynomials, the degree of the polynomial, and others. In this article, we will learn about, Polynomials, Polynomial Equations, Solving Cubic Equations Or how to solve cubic equations, and others in detail.
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