Answer :
To determine the statement that is true according to the Rational Root Theorem for the polynomial [tex]\( f(x) = 25x^5 - x - 5x^4 - x^0 - 49 \)[/tex], we need to understand the Rational Root Theorem.
The Rational Root Theorem states that any potential rational root of the polynomial [tex]\( f(x) \)[/tex], which can be written as a fraction [tex]\(\frac{p}{q}\)[/tex], must satisfy the following conditions:
- [tex]\( p \)[/tex] (the numerator) must be a factor of the constant term (the term without [tex]\( x \)[/tex]), and
- [tex]\( q \)[/tex] (the denominator) must be a factor of the leading coefficient (the coefficient of the term with the highest power of [tex]\( x \)[/tex]).
Given the polynomial [tex]\( f(x) = 25x^5 - x - 5x^4 - x^0 - 49 \)[/tex], we identify:
- The constant term, [tex]\( c \)[/tex], is [tex]\(-49\)[/tex],
- The leading coefficient, [tex]\( a_n \)[/tex], is [tex]\( 25 \)[/tex].
### Step-by-step explanation:
1. Identify Factors
- Factors of the constant term ([tex]\( -49 \)[/tex]): [tex]\( \pm 1, \pm 7, \pm 49 \)[/tex].
- Factors of the leading coefficient ([tex]\( 25 \)[/tex]): [tex]\( \pm 1, \pm 5, \pm 25 \)[/tex].
2. Formulate Possible Rational Roots
- According to the theorem, any rational root [tex]\(\frac{p}{q}\)[/tex] must be such that [tex]\( p \)[/tex] (numerator) is a factor of [tex]\(-49\)[/tex], and [tex]\( q \)[/tex] (denominator) is a factor of [tex]\( 25 \)[/tex].
3. List Possible Rational Roots
- The possible rational roots could be [tex]\(\pm \frac{1}{1}, \pm \frac{1}{5}, \pm \frac{1}{25}, \pm \frac{7}{1}, \pm \frac{7}{5}, \pm \frac{7}{25}, \pm \frac{49}{1}, \pm \frac{49}{5}, \pm \frac{49}{25} \)[/tex].
From these steps, we conclude:
- Any rational root of the polynomial [tex]\( f(x) \)[/tex] is a factor of the constant term ([tex]\(-49\)[/tex]) divided by a factor of the leading coefficient ([tex]\(25\)[/tex]).
Thus, the correct statement is:
- Any rational root of [tex]\( f(x) \)[/tex] is a factor of [tex]\(-49\)[/tex] divided by a factor of [tex]\(25\)[/tex].
So, the statement that is true according to the Rational Root Theorem is:
Any rational root of [tex]\( f(x) \)[/tex] is a factor of -49 divided by a factor of 25.
The Rational Root Theorem states that any potential rational root of the polynomial [tex]\( f(x) \)[/tex], which can be written as a fraction [tex]\(\frac{p}{q}\)[/tex], must satisfy the following conditions:
- [tex]\( p \)[/tex] (the numerator) must be a factor of the constant term (the term without [tex]\( x \)[/tex]), and
- [tex]\( q \)[/tex] (the denominator) must be a factor of the leading coefficient (the coefficient of the term with the highest power of [tex]\( x \)[/tex]).
Given the polynomial [tex]\( f(x) = 25x^5 - x - 5x^4 - x^0 - 49 \)[/tex], we identify:
- The constant term, [tex]\( c \)[/tex], is [tex]\(-49\)[/tex],
- The leading coefficient, [tex]\( a_n \)[/tex], is [tex]\( 25 \)[/tex].
### Step-by-step explanation:
1. Identify Factors
- Factors of the constant term ([tex]\( -49 \)[/tex]): [tex]\( \pm 1, \pm 7, \pm 49 \)[/tex].
- Factors of the leading coefficient ([tex]\( 25 \)[/tex]): [tex]\( \pm 1, \pm 5, \pm 25 \)[/tex].
2. Formulate Possible Rational Roots
- According to the theorem, any rational root [tex]\(\frac{p}{q}\)[/tex] must be such that [tex]\( p \)[/tex] (numerator) is a factor of [tex]\(-49\)[/tex], and [tex]\( q \)[/tex] (denominator) is a factor of [tex]\( 25 \)[/tex].
3. List Possible Rational Roots
- The possible rational roots could be [tex]\(\pm \frac{1}{1}, \pm \frac{1}{5}, \pm \frac{1}{25}, \pm \frac{7}{1}, \pm \frac{7}{5}, \pm \frac{7}{25}, \pm \frac{49}{1}, \pm \frac{49}{5}, \pm \frac{49}{25} \)[/tex].
From these steps, we conclude:
- Any rational root of the polynomial [tex]\( f(x) \)[/tex] is a factor of the constant term ([tex]\(-49\)[/tex]) divided by a factor of the leading coefficient ([tex]\(25\)[/tex]).
Thus, the correct statement is:
- Any rational root of [tex]\( f(x) \)[/tex] is a factor of [tex]\(-49\)[/tex] divided by a factor of [tex]\(25\)[/tex].
So, the statement that is true according to the Rational Root Theorem is:
Any rational root of [tex]\( f(x) \)[/tex] is a factor of -49 divided by a factor of 25.
