Questions: According to the Rational Roots Theorem, which statement about f(x)=25 x^7-x^6-5 x^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 factor of 25 divided Any rational root of f(x) is a multiple of 25 by a factor of -49. divided by a multiple of -49. Any rational root of f(x) is a factor of -49 divided by a factor of 25.

According to the Rational Roots Theorem, which statement about f(x)=25 x^7-x^6-5 x^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 factor of 25 divided

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

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

Transcript text: According to the Rational Roots Theorem, which statement about $f(x)=25 x^{7}-x^{6}-5 x^{4}+x-49$ 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 factor of 25 divided Any rational root of $f(x)$ is a multiple of 25 by a factor of -49 . divided by a multiple of -49. Any rational root of $f(x)$. is a factor of -49 divided by a factor of 25.

Solution

Solution Steps

To determine which statement about the polynomial $ f(x) = 25x^7 - x^6 - 5x^4 + x - 49 $ is true according to the Rational Roots Theorem, we need to identify the possible rational roots. The Rational Roots Theorem states that any rational root, expressed as a fraction $\frac{p}{q}$, has $p$ as a factor of the constant term and $q$ as a factor of the leading coefficient. Here, the constant term is $-49$ and the leading coefficient is $25$. Therefore, the possible rational roots are the factors of $-49$ divided by the factors of $25$.

Step 1: Identify the Polynomial

We are given the polynomial $ f(x) = 25x^7 - x^6 - 5x^4 + x - 49 $. To analyze its rational roots, we will apply the Rational Roots Theorem.

Step 2: Determine Factors

According to the Rational Roots Theorem, any rational root $ \frac{p}{q} $ must have $ p $ as a factor of the constant term and $ q $ as a factor of the leading coefficient. Here, the constant term is $-49$ and the leading coefficient is $25$.

Factors of $-49$: $ \{-1, -7, -49, 1, 7, 49\} $
Factors of $25$: $ \{-1, -5, -25, 1, 5, 25\} $

Step 3: Calculate Possible Rational Roots

Using the factors identified, we calculate the possible rational roots by forming the fractions $ \frac{p}{q} $ where $ p $ is a factor of $-49$ and $ q $ is a factor of $25$. The resulting possible rational roots are:

\[ \begin{align_} \frac{-1}{1} &= -1 \\ \frac{-1}{5} &= -0.2 \\ \frac{-1}{25} &= -0.04 \\ \frac{-7}{1} &= -7 \\ \frac{-7}{5} &= -1.4 \\ \frac{-7}{25} &= -0.28 \\ \frac{-49}{1} &= -49 \\ \frac{-49}{5} &= -9.8 \\ \frac{-49}{25} &= -1.96 \\ \frac{1}{1} &= 1 \\ \frac{1}{5} &= 0.2 \\ \frac{1}{25} &= 0.04 \\ \frac{7}{1} &= 7 \\ \frac{7}{5} &= 1.4 \\ \frac{7}{25} &= 0.28 \\ \frac{49}{1} &= 49 \\ \frac{49}{5} &= 9.8 \\ \frac{49}{25} &= 1.96 \\ \end{align_} \]

After calculating, the unique possible rational roots are: \[ \{ -49, -9.8, -7, -1.96, -1.4, -1, -0.28, -0.2, -0.04, 0.04, 0.2, 1, 1.4, 7, 9.8, 49 \} \]

Step 4: Analyze the Statements

Now we analyze the provided statements based on the Rational Roots Theorem:

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