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

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 \frac{\text{Factors of }a_0}{\text{Factors of }a_n}`

In this case,

`a_(0)=-49` and
`a_(n)=25`

So,

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