Question

Which are the real zeros of this function?
f(x) = x3 – 6x2 – 16x

Answer 1

To find the zeros of the function
`f(x)= x^(3) -6x^(2)-16x` we need to factorize the expression.


`f(x)= x^(3) -6x^(2)-16x =x(x^(2) -6x-16 )`

a nice way to factorize
`x^(2) -6x-16`, if possible, is by completing the square as follows:


`x^(2) -6x-16 =x^(2) -2*3x+ (3)^(2)-(3)^(2) -16`


`=(x^(2) -2*3x+ (3)^(2))-9-16= (x-3)^(2)-25= (x-3)^(2)- 5^(2)`

now we use the difference of squares formula
`a^(2) - b^(2) =(a+b)(a-b)`:


`(x-3)^(2)- 5^(2)=[(x-3)+5][(x-3)-5]=[x+2][x-8]`


finally, we combine the results:


`f(x)=x(x+2)(x-8)`

the zeros of f, are the values of x which make f(x)=0,

they are x=0, x=-2 and x=8


Answer: {-2, 0, 8}