f(x)=x^4-6x^3+8x^2-21

Question

list all possible zeros of the function using rational zero theorem

Answer 1

To use the rational roots theorem, we first need to make sure our terms are combined and arranged in such a way that we can find the constant term (the one with no variable) and the term of the highest degree.

In this case, the work is already done. x⁴ is the term of the highest degree and -21 is the constant term.

The coefficient of x⁴ is 1.

Now we find all the factors of 1 and -21

The factors of -21 are ±1, ±3, ±7, or ±21

The factors of 1 are ±1

 

You're pretty much done, you just have to combine these factors to make your possible rational roots. The factors of your constant term are all the possible numerators, and the factors of your leading term are all the possible denominator.

Once again, 

Rational roots: (factors of constant term)/(factors of leading coefficient)

In this case our possible rational roots are (±1, ±3, ±7, or ±21)/(±1)

This simplifies to ±1, ±3, ±7, or ±21

 

I hope this helps! Please ask if this isn't clear