Q. Please help me on "HOW TO SOLVE CUBIC POLYNOMIALS" BY SPLITTING THE MIDDLE TERM. THANK YOU!
Hi Kiruba, Good Question.
Here is the concept and method to solve these type of questions.
Topic: Algebra, Rational / Polynomials
Concept : Cubic polynomials can be solved by factor method and splitting middle term. Let's understand with an example.
Example: Take a cubic polynomial as x3 - 2x2 - x + 2
Method : Here are the steps and explanation.
- Let p(x) = x3 - 2x2 - x + 2
- Here the constant term is 2 (as shown in red colour).
- So write factors for 2
- Factors for 2 are : +1, -1, +2, -2
- Now let's substitute each value and see if it satisifies the p(x) or not.
- p(1) = (1)3 - 2*(1)2 - 1 + 2
On solving we get p(1) = 0
- So x = 1 satisfies the p(x) hence (x-1) is one factor of p(x). Right?
- So we got one factor and need to find remaining two factors.
- One way is substitute other values as in step 4 and if they satisfies the p(x) then you get other two factors too. Else use splitting middle term method as shown below.
- Write p(x) by splitting terms such that you get (x-1) as common.
- Rewriting p(x) = x3 - x2 - x2 + x - 2x + 2
- Taking (x-1) common we get p(x) = x2(x-1) - x(x-1) - 2(x-1)
- On solving, p(x) = (x-1)(x2 - x - 2)
- So now we got quadratic equation (x2 - x - 2) which can be solved by splitting middle term easily.
- Middle term in (x2 - x - 2) is -x which can be splitted as -2x and +x
- So (x2 - x - 2) can be written as (x2 - 2x + x - 2)
- Taking common out, we get x(x- 2) + 1 (x - 2)
- On solving, we get (x- 2)(x+1) as other two factors.
- So all three factors of p(x) are (x-1)(x- 2)(x+1)
Kiruba, hope it helped.Back to blog