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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.

Verified Answer :

Hagrid

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:

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

Step-by-step 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

\(\pm \frac{factors of a_0}{factors of a_n}\)

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 \(a_n=25\)


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