Question

How do you find other zeros given one zero \[f(x) = {x^3} - 8{x^2} + 4x + 48;4\]?

Answer 1

Hint: The given question hints us towards that the question has itself given one zero which means it has given one factor already and that we have to find the other factors of the given polynomial equation. The given polynomial equation has one factor which is $ x - 4 = 0 $ and we have to find other factors of the equation. To find the other factors of the equation we will divide the given polynomial equation by the one factor which is given to us and then factorize the quotient.

Complete step-by-step answer:
The given question gives us one factor which is
 $ x - 4 = 0 $
And we have to find the other factors which we will do in the following steps by division of the given polynomial equation.
We divide $ \dfrac{{{x^3} - 8{x^2} + 4x + 48}}{{x + 4}} $
Which gives us $ {x^3} - 8{x^2} + 4x + 48 = (x - 4)({x^2} - 4x - 12) $
The remaining quadratic expression obtained can be broken down into other factors as
 $ (x - 6)(x + 2) $
The above equation thus becomes $ {x^3} - 8{x^2} + 4x + 48 = (x - 4)(x - 6)(x + 2) $
Thus the following equations is solved the other zeroes as asked in the given question are as follows:
 $ x - 6 = 0 $
Gives one zero as $ x = 6 $ and for the other zero it is given as
 $ x + 2 = 0 $ which upon solving gives us
 $ x = - 2 $
Thus all the zeroes of the given polynomial expression are :
  $ 4,6, - 2 $
So, the correct answer is “ $ 4,6, - 2 $ ”.

Note: The above numbers are called as zeroes of the above expression due to the fact that they follow remainder equation and are their factors so if we put these values of zeroes in our expression we will get remainder as zero for the above given expression.