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According to the Rational Roots Theorem, which statement about f(x) = 25x7 – x6 – 5x4 + x – 49 is true?


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

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

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

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

Verified Answer :

Aliwohaish12

C is the correct answer :-)

InesWalston

Answer-

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

Solution-

The given polynomial is

\(25x^7-x^6-5x^4+x-49\)

Rational Root Theorem-

\(a_{n}x^{n}+a_{n-1}x^{n-1}+\cdots +a_{0}=0\)

If \(a_{0}\) and \(a_{n}\) are nonzero, then each rational solution x will be,

\(x=\pm \dfrac{\text{Factors of }a_0}{\text{Factors of }a_n}\)

In this case,

\(a_{0}=-49\) and \(a_{n}=25\)

So,

\(x=\pm \dfrac{\text{Factors of }(-49)}{\text{Factors of }(25)}\)

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