Question

How do you factor \[2{x^3} + 7{x^2} - 3x - 18\] ?

Answer 1

Hint: Here in this question, the given equation in the form of a cubic equation with x has a variable. This equation can be solved by using factor theorem or synthetic division. First find the one root by a guesswork or trial and error method and write the quadratic equation by using the method of synthetic equation and further simplify the quadratic equation by factorization for finding other two roots.

Complete step-by-step answer:
A cubic function or equation is a third-degree polynomial. A general polynomial function has the form:
 \[f(x) = a{x^n} + b{x^{n - 1}} + c{x^{n - 2}}...v{x^3} + w{x^2} + zx + k\]
Here, x is the variable, n is simply any number (and the degree of the polynomial), k is a constant and the other elements or letters are constant coefficients for each power of x. so a cubic function has n=3, and is simply: \[f(x) = a{x^3} + b{x^2} + c{x^1} + d\]
The easiest way to solve a cubic equation involves a bit of guesswork and an algorithmic type of process called synthetic division. Though it is basically the same as the trial-and-error method for cubic equation solutions.
The fundamental theorem of algebra is defined as that every polynomial has at least one zero in the complex numbers. So F(x)=0.
Consider the given cubic equation and we will compute it to zero
 \[ \Rightarrow \,\,\,2{x^3} + 7{x^2} - 3x - 18 = 0\]
Try to find one of the roots by guessing.
You have to guess one of the values of x, let the first factor be \[x = - 2\] , then
 \[ \Rightarrow \,\,2\,{( - 2)^3} + 7{\left( { - 2} \right)^2} - 3( - 2) - 18 = 0\]
 \[ \Rightarrow \,\,\,2( - 8) + 7(4) - 3( - 2) - 18 = 0\]
 \[
   \Rightarrow \,\,\, - 16 + 28 + 6 - 18 = 0 \\
   \Rightarrow 0 = 0 \;
 \]
This means x=-2 is a root of the cubic equation. You can get the answer without much thought and it is also time-consuming (especially if you have to go to higher factors before finding a root). Luckily, when you’ve found one root, you can solve the rest of the equation easily.
The key is incorporating the factor theorem. This states that if x = s is a solution, then (x – s) is a factor that can be pulled out of the equation. For this situation, s = -2, and so (x+2) is a factor.
Now when we divide the given equation \[2{x^3} + 7{x^2} - 3x - 18\] by \[x + 2\] , we get
 \[ \Rightarrow 2{x^3} + 7{x^2} - 3x - 18 = \left( {x + 2} \right)\left( {2{x^2} + 3x - 9} \right)\]
Again we can factorise the second term.
 \[ \Rightarrow \left( {2{x^2} + 3x - 9} \right) = 2{x^2} + 6x - 3x - 9\]
Now take the common terms and rewrite
 \[ \Rightarrow \left( {2{x^2} + 3x - 9} \right) = 2x(x + 3) - 3(x + 3)\]
 \[ \Rightarrow \left( {2{x^2} + 3x - 9} \right) = (2x - 3)(x + 3)\]
Therefore now we have to factorise the second term.
Therefore the factors of the given equation \[ \Rightarrow 2{x^3} + 7{x^2} - 3x - 18 = \left( {x + 2} \right)(2x - 3)(x + 3)\]
So, the correct answer is “ \[ \Rightarrow 2{x^3} + 7{x^2} - 3x - 18 = \left( {x + 2} \right)(2x - 3)(x + 3)\] ”.

Note: The cubic equation contains 3 roots. To find or solve the cubic equation, solved by using the synthetic division method. The one value or root of the given equation determined by the synthetic division. Then the quadratic equation will remain, to the quadratic equation the sum product rule or by standard formula is applied to determine the other two roots.