**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 *x ^{3 }- 2x^{2} - x + 2*

**Method : **Here are the steps and explanation.

- Let p(x) =
*x*^{3 }- 2x^{2}- x +*2* - Here the constant term is
(as shown in red colour).*2* - So write factors for
*2* - Factors for
are :*2**+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) =
*x*^{3 }*- x*^{2 }- x^{2}+ x - 2x*+ 2* - Taking (x-1) common we get p(x) =
*x*^{2}(x-1) - x(x-1) - 2(x-1) - On solving,
*(x-1)(x*^{2}- x - 2) - So now we got quadratic equation
*(x*which can be solved by splitting middle term easily.^{2}- x - 2) - Middle term in
*(x*is^{2}- x - 2)*-x*which can be splitted as*-2x and +x* - So
*(x*can be written as^{2}- x - 2)*(x*^{2}- 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.

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