Respuesta :
Answer:
[tex]x=-2,0,3[/tex]
Step-by-step explanation:
We have been given a function [tex]f(x)=15x^3-15x^2-90x[/tex]. We are asked to find the zeros of our given function.
To find the zeros of our given function, we will equate our given function by 0 as shown below:
[tex]15x^3-15x^2-90x=0[/tex]
Now, we will factor our equation. We can see that all terms of our equation a common factor that is [tex]15x[/tex].
Upon factoring out [tex]15x[/tex], we will get:
[tex]15x(x^2-x-6)=0[/tex]
Now, we will split the middle term of our equation into parts, whose sum is [tex]-1[/tex] and whose product is [tex]-6[/tex]. We know such two numbers are [tex]-3\text{ and }2[/tex].
[tex]15x(x^2-3x+2x-6)=0[/tex]
[tex]15x((x^2-3x)+(2x-6))=0[/tex]
[tex]15x(x(x-3)+2(x-3))=0[/tex]
[tex]15x(x-3)(x+2)=0[/tex]
Now, we will use zero product property to find the zeros of our given function.
[tex]15x=0\text{ (or) }(x-3)=0\text{ (or) }(x+2)=0[/tex]
[tex]15x=0\text{ (or) }x-3=0\text{ (or) }x+2=0[/tex]
[tex]\frac{15x}{15}=\frac{0}{15}\text{ (or) }x-3=0\text{ (or) }x+2=0[/tex]
[tex]x=0\text{ (or) }x=3\text{ (or) }x=-2[/tex]
Therefore, the zeros of our given function are [tex]x=-2,0,3[/tex].
