How to solve Cubic Polynomial

In this post I'll keep on explaining, How To solve cubic polynomials ?

Earlier in my previous post click here for video I've shared a video in which a short trick explained to solve Cubic Polynomial within 1 minute or less.

Now I'm going to explain the exact method to solve these kinds of Polynomials, so read this post till end for better understanding.

Cubic polynomial

• A standard form of cubic polynomial :

P(x) = ax ³ + bx ² + cx + d.If a polynomial given
P(x) = x³ - 4x² - 7x + 10

• I want to clear the major concept of polynomial,

•   Number of Factors of any polynomial is equal to the highest exponents of polynomial (or degree of polynomial).

that means for cubic polynomial there must be 3 factors exist as the degree of polynomial is 3.
Also read

• Completing the square method
• Quadratic formula

so, among these 3 factors we have to predict or put a guess for the first factor (this may be annoying thing but trust me this method will never let's you down, so, be with me)

so for above equation I guess it may be one out of among 1, -1, 2, -2, 3, -3 (enough).

so one by one put these factor into given polynomial, the one (factor) who make the polynomial is equal to zero that particular value will be our first factor.
so for 1,

we put x = 1
in
P(x) = x³ - 4x² - 7x + 10

p(1) = 1³ - 4×1² - 7× 1 + 10
        = 1 - 4 - 7+ 10
        = 11 - 11
        = 0

voila..!!
we've found in one trial. Now we have our first factor i.e. x = 1
(x-1).     ....................................(i)


now for other two factors we have to perform long Division method.

= [ x³ - 4x² - 7x + 10] / (x -1)
yields

= x² - 3x - 10.


now we have one Quadratic equation i.e.

x² - 3x - 10.

for obtaining those two factors we've to factorise this using Middle term splitting.

= x² - 3x - 10
=  x² - 5x  + 2x - 10
= x ( x - 5 ) + 2 ( x - 5)
= (x + 2) ( x - 5)

finally we have all the three factors i.e.
(x -1 ) from equation ...(i) and the other two ( x- 2),(x-5).


x³ - 4x² - 7x + 10 = (x-1)(x+2)(x-5)

so, we've reached to end of the topic cubic polynomial no doubt there can be other methods as well but I personally found that this method is little lengthy rather this will be easier to solve any cubic polynomial.

If you have any doubts and Queries, I would like to hear from you.
Keep Learning.....!!!