Student: KIRUBA

Q. Please help me on "HOW TO SOLVE CUBIC POLYNOMIALS" BY SPLITTING THE MIDDLE TERM. THANK YOU!


Answer :

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. yes

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