Respuesta :
We are given the function
[tex]f(x)=5x\cdot(x-7)^2\cdot(x-16)^2[/tex]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
[tex]5x\cdot(x-7)^2\cdot(x-16)^2=0[/tex]Note that in this case the function f is a product of the following functions
[tex]5x[/tex][tex](x-7)^2[/tex]and
[tex](x-16)^2[/tex]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
[tex]5x=0[/tex]By dividing both sides by 5, we get
[tex]x=\frac{0}{5}=0[/tex]so one zero is x=0.
Function (x-7)²:
We have the following equation:
[tex](x-7)^2=0[/tex]Recall that the square of a number can be zero if and only if the number itself is zero. So we get
[tex]x\text{ -7=0}[/tex]By adding 7 on both sides, we get
[tex]x=0+7=7[/tex]So another zero of the function is x=7.
Function (x-16)²:
We have the following equation:
[tex](x-16)^2=0[/tex]Recall tha the square of a number can be zero if and only if the number itself is zero. So we get
[tex]x\text{ - 16=0}[/tex]By adding 16 on both sides, we get
[tex]x=0+16=16[/tex]so the last zero is 16.
In conclusion, the zeros of the function f are x=0, x=7 and x=16.
