We are given the polynomial equation shown below.
x^3-4x^2-9x+36=0
There are different methods we can use to solve this equation. We will illustrate how to do this by factoring and by graphing.
Solving the Equation by Factoring
Notice that there is no common factor to all of the polynomial's terms. However, if we group them, we can find a common monomial factor for each group.
x^3-4x^2 -9x+36 & = x^2(x-4) + 9(4-x)
Notice that we can rewrite the second term from the right-hand side as -9(x-4). By doing this, we can identify the common factor (x-4) appearing in each term of the right hand side. Therefore, we can factor out this binomial.
x^2 (x-4) -9 (x-4)
⇕
(x^2-9) (x-4)
Now we can identify the term (x^2-9) as having the difference of two squares pattern. Hence, it can be factored as the product of two conjugate binomials, completely factoring the original polynomial.
(x^2-9)(x-4)
⇕
(x-3)(x+3)(x-4)
We have successfully factored the original polynomial equation.
x^3-4x^2-9x+36=0
⇕
(x-3)(x+3)(x-4)=0
Finally, by referring to the Zero Product Property we can state its roots to be x= 3, x=-3 and x=4. Note that this method requires several algebraic process, but if we can take the polynomial to its factored form, then we can identify the exact solutions.
Solving the Equation by Graphing
Another way to solve the equation is by graphing the related function f(x) = x^3-4x^2-9x+36. Notice that we recover the original equation when f(x)= 0.
f(x)= 0 ⇔ x^3-4x^2-9x+36 = 0
Since the points on the graph have the form (x,f(x)) the solutions will be the x-intercepts, since they have the form (x,0).
From the graph above we can identify the roots as x=3, x=-3 and x=4. Notice that this method might require the use of a graphic calculator, which we may not always have at hand. Furthermore, if the roots are not exact values identifying them can be complicated.
Conclusions
Although each method has its benefits and drawbacks, for this case graphing is much more convenient, especially since all of the roots are integer numbers.
