Question

According to the Rational Roots Theorem, which statement about f(x) = 25x^7 – x^6 – 5x^4 + x – 49 is true?

Any rational root of f(x) is a multiple of –49 divided by a multiple of 25.

Any rational root of f(x) is a multiple of 25 divided by a multiple of –49.

Any rational root of f(x) is a factor of –49 divided by a factor of 25.

Any rational root of f(x) is a factor of 25 divided by a factor of –49.

Answer 1

According to the Rational Roots Theorem, the statement about f(x) = 25x^7 – x^6 – 5x^4 + x – 49 which is true is:

Any rational root of f(x) is a factor of –49 divided by a factor of 25.

The key points are "factors", and "ratio between the constant term and the coefficient of the highest order (exponent) term"

Answer 2

Any rational root of f(x) is a factor of -49 divided by a factor of 25.

Explanation:

The Rational Root Theorems states that :

If the polynomial
`P(x)= a _nx^n +{a_(n-1)x}^(n-1)+............{a_(2)x}^(2)+{a_(1)x}^(1)+a_0` has any rational roots, then they must be in the form of

`\pm (factors of a_0)/(factors of a_n)`

Consider the polynomial

`f(x)=25x^7-x^6-5x^4+x-49`

in this case, we have
`a_0=-49` and
`a_n=25`

Any Rational root of f(x) is a factor of
`a_0=-49` divided by a factor of
`a_n=25`