Question

Consider the function f(x) = x^3 - 7x^2 - 2x + 14

What is the total number of zeros of f(x)? Explain how you got your answer.
Find the zeros of f(x). Explain your procedure as you solved it.

Answer 1

There are 3 zeros, which are:

`x=7,\:x=-√(2),\:x=√(2)`

Explanation:

Given the function

`f\left(x\right)\:=\:x^3\:-\:7x^2\:-\:2x\:+\:14`

To get the zeros of
`f(x)`, set
`y` or
`f(x) =0`

so

`0=x^3-7x^2-2x+14`

as

`x^3\:-\:7x^2\:-\:2x\:+\:14=\left(x-7\right)\left(x+√(2)\right)\left(x-√(2)\right)`

so

`\left(x-7\right)\left(x+√(2)\right)\left(x-√(2)\right)=0`

Using the zero factor principle:

`\mathrm{ \:If}\:ab=0\:\mathrm{then}\:a=0\:\mathrm{or}\:b=0\:\left(\mathrm{or\:both}\:a=0\:\mathrm{and}\:b=0\right)`

so

`x-7=0\quad \mathrm{or}\quad \:x+√(2)=0\quad \mathrm{or}\quad \:x-√(2)=0`

solving

`x-7=0`

• 
  `x=7`

`x+√(2)=0`

• 
  `x=-√(2)`

`x-√(2)=0`

• 
  `x=√(2)`

Therefore, there are 3 zeros, which are:

`x=7,\:x=-√(2),\:x=√(2)`