Question

Find all zeros of the function f(x)=5x(x-7)^2(x-16)^2

Answer 1

We are given the function

`f(x)=5x\cdot(x-7)^2\cdot(x-16)^2`

we want to find all zeros of this function. That is, we want to find all value of x such that f(x) is zero. So, we have the following equation

`5x\cdot(x-7)^2\cdot(x-16)^2=0`

Note that in this case the function f is a product of the following functions

`5x`
`(x-7)^2`

and

`(x-16)^2`

so, since f is the product of these functions, for f to have the value of 0, at least one of this functions should be 0.

So we analyze each function separately.

Function 5x:

We have the following equation

`5x=0`

By dividing both sides by 5, we get

`x=(0)/(5)=0`

so one zero is x=0.

Function (x-7)²:

We have the following equation:

`(x-7)^2=0`

Recall that the square of a number can be zero if and only if the number itself is zero. So we get

`x\text{ -7=0}`

By adding 7 on both sides, we get

`x=0+7=7`

So another zero of the function is x=7.

Function (x-16)²:

We have the following equation:

`(x-16)^2=0`

Recall tha the square of a number can be zero if and only if the number itself is zero. So we get

`x\text{ - 16=0}`

By adding 16 on both sides, we get

`x=0+16=16`

so the last zero is 16.

In conclusion, the zeros of the function f are x=0, x=7 and x=16.