Answer:
First Option is correct.
(x+2)(x+3)(x-4)
Step-by-step explanation:
Given:
The given factor is.
[tex]f(x)=x^{3}+x^{2}-14x-24[/tex]
We need to find the factors of given factor.
Solution:
[tex]f(x)=x^{3}+x^{2}-14x-24[/tex]
Substitute (-4x-10x) in the place of -14x
[tex]f(x)=x^{3}+x^{2}-4x-10x-24[/tex]
Rearrange the equation:
[tex]f(x)=(x^{3}-4x)+(x^{2}-10x-24)[/tex]
Now we factorised the above equation.
The factors of [tex]x^{2} -10x-24=(x+2)(x-12)[/tex]
Now we substitute (x+2)(x-12) in the place of [tex]x^{2} -10x-24[/tex]
And in the place of [tex]x^{3}-4x=x(x^{2}-4)[/tex]
[tex]f(x)=x(x^{2}-4)+(x+2)(x-12)[/tex]
Simplify [tex]x^{2} -4=x^{2} -(2)^{2} = (x+2)(x-2)[/tex]
[tex]f(x)=x[(x+2)(x-2)]+(x+2)(x-12)[/tex]
Common factor for above function (x+2)
[tex]f(x)=(x+2)[x(x-2)+(x-12)][/tex]
Simplify.
[tex]f(x)=(x+2)[x^{2}-2x+x-12][/tex]
[tex]f(x)=(x+2)(x^{2}-x-12)[/tex]
The factor of [tex]x^{2}-x-12=x^{2}-4x+3x-12=x(x-4)+3(x-4)=(x-4)(x+3)[/tex]
So the factors of the function.
[tex]f(x)=(x+2)(x-4)(x+3)[/tex]
Therefore, the factors of the given function are (x+2)(x-4)(x+3)
