Question

List all possible rational zeros are given by the rational zeros theorem (but don't check to see which actually are zeros). (enter your answers as a comma-separated list.)

p(x)=x3−7x2+19

Answer 1

All possible rational zeros are given by the rational zeros theorem are 1, -1, 19 and -19.

Explanation:

The rational zeros theorem indicates that for a polynomial

f (x) = aₙxⁿ + aₙ₋₁xⁿ⁻ + ⋯ + a₂x² + a₁x + a₀

where aₙ, aₙ₋₁, ⋯ a₀ are integers, the rational roots can be determined from of the factors of aₙ and a₀.

Being p divisor of the independent term a₀ and q divisor of the coefficient principal aₙ, then all rational factors or possible rational roots will take the form of
`(p)/(q)`

In the case of p(x)=x³ -7x² +19, aₙ=1 and a₀=19

So:

Dividers of the independent term (19): p=±1,±19

Divisors of the main coefficient (1): q=±1

Being the possible roots of the polynomial:
`(p)/(q)=`±1, ±19

All possible rational zeros are given by the rational zeros theorem are 1, -1, 19 and -19.