Answer:
x=2 ; x=-4 or x=-3/2
Step-by-step explanation:
We are given a cubic polynomial
[tex]2x^3+7x^2-10x-24=0[/tex]
Here as we are having a cubic equation to factorize, we have to find some value of x such that when we put that value in place of x , it gives us a 0. We have to try it randomly like first we try it for x=1 then x=2 and so on.
On trying for x=2 , we see that the equation becomes 0=0 and hence x=2 is one of the solution. And also as x=2 is one of the solution, (x-2) must be one of the factor. Now we use this property to determine the other factor like this.
[tex]2x^3+7x^2-10x-24=0[/tex]
Adding and subtracing [tex]4x^2[/tex] to [tex]2x^3[/tex] we get
[tex]2x^3-4x^2+4x^2+7x^2-10x-24=0[/tex]
Now we take out [tex]2x^2[/tex] as GCF
[tex]2x^2(x-2)+11x^2-10x-24=0[/tex]
Now subtracting and adding [tex]22x[/tex] to [tex]11x^2[/tex]
[tex]2x^2(x-2)+11x^2-22x+22x-10x-24=0[/tex]
taking [tex]11x^2[/tex] as GCF out
[tex]2x^2(x-2)+11x(x-2)+22x-10x-24=0[/tex]
[tex]2x^2(x-2)+11x(x-2)+12x-24=0[/tex]
[tex]2x^2(x-2)+11x(x-2)+12(x-2)=0[/tex]
Now taking [tex](x-2)[/tex] out as GCF
[tex](x-2)(2x^2+11x+12)[/tex]
Now we need to split the middle term of [tex](2x^2+11x+12)[/tex] to factorize it in such a way that their product is 2*12 and sum is 11. We have 8 and 3 as these factors.
Hence it can be factored like this
[tex](2x^2+11x+12)\\2x^2+8x+3x+12\\2x(x+4)+3(x+4)\\(2x+3)(x+4)[/tex]
Hence our answer will be
[tex](x-2)(2x+3)(x+4)=0[/tex]
And thus the solutions will be
x-2=0 : x=2
x+4=0 ; x=-4
2x+3=0 ; x=-[tex]\frac{3}{2}[/tex]
