Question

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.

Answer 1

C is the correct answer :-)

Answer 2

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,

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