If (x²+1) (y³+1) (z^4+1) = 0, which of the following must equal 0?

Question

A. x+1

B. y+1

C z+1

D. x+y+z

E. xyz

 

This was heard. Please help. Thank you!

Answer 1

If y+1=0, then y=-1 and then y³ = -1, so the answer is "B".

The other terms have even exponents which always give positive values of the term.

Answer 2

For  (x²+1) (y³+1) (z^4+1) = 0,

  then either

        (x²+1) = 0         [eq1]

or     (y³+1) = 0         [eq2]

or     (z^4+1) = 0       [eq3]

 

Now, for either eq1 or eq3 to be true, a number to an even power must = -1.   This is not true for negative real numbers, for zero, or for positive real numbers. This only happens for imaginary numbers.

 

However, y to an odd power (3) may produce -1  [specifically if y=-1] 

          [you do not need to factor this to realize that it is true, so y+1=0 ]

Answer 3

For the product in the above equation to be equal to zero at least one of the factors must be equal to zero.

 

The factors in the equation are not listed as possible answers, so you will need to try to factor the factors.

 

Do you know which of those factors in the equations can be broken down into smaller factors?