Rational Root Theorem: Find The Polynomial With $-\frac{7}{8}$ As A Potential Root

Hey math enthusiasts! Today, we're diving into a super useful tool in algebra called the Rational Root Theorem. This theorem is your best friend when you're trying to find possible rational roots (solutions) of a polynomial equation. It helps narrow down the endless possibilities to a manageable list. We'll be tackling a specific problem: According to the Rational Root Theorem, $-\frac{7}{8}$ is a potential rational root of which function? We've got three functions to choose from, and we need to use our trusty theorem to figure out which one fits the bill. So, grab your calculators, dust off your notebooks, and let's get ready to crunch some numbers!

Understanding the Rational Root Theorem

The Rational Root Theorem is a fundamental concept in algebra that provides a systematic way to identify potential rational roots of a polynomial equation with integer coefficients. Let's break it down. Imagine you have a polynomial like $P(x) = a_n x^n + a_{n-1} x^{n-1} + \dots + a_1 x + a_0$, where all the coefficients ($a_n, a_{n-1}, \dots, a_0$) are integers. The Rational Root Theorem states that if this polynomial has a rational root, it can be expressed in the form $\frac{p}{q}$, where $p$ is an integer factor of the constant term ($a_0$) and $q$ is an integer factor of the leading coefficient ($a_n$). In simpler terms, any rational root of the polynomial must be a fraction formed by dividing a factor of the last number (the constant term) by a factor of the first number (the coefficient of the highest power of $x$). This theorem doesn't tell you which of these potential roots are actual roots, but it gives you a finite list to test. This is incredibly powerful because, without it, you might be testing random fractions forever! The theorem is particularly helpful for polynomials of degree 3 or higher, where finding roots can become quite challenging. It's like having a map that shows you all the possible treasure locations, even if you don't know the exact spot yet. Remember, $p$ must be a divisor of $a_0$, and $q$ must be a divisor of $a_n$. Both $p$ and $q$ can be positive or negative. So, the set of all possible rational roots is the set of all possible $\frac{p}{q}$ values.

Applying the Theorem to Our Problem

Now, let's apply the Rational Root Theorem to our specific problem. We are given a potential rational root, $-\frac{7}{8}$, and we need to determine which of the following functions it could be a root of:

1. $f(x)=24 x^7+3 x^6+4 x^3-x-28$
2. $f(x)=28 x^7+3 x^6+4 x^3-x-24$
3. $f(x)=30 x^7+3 x^6+4 x^3-x-56$

According to the Rational Root Theorem, if $-\frac{7}{8}$ is a rational root of a polynomial function $f(x) = a_n x^n + \dots + a_0$, then the numerator, $-7$, must be a factor of the constant term ($a_0$), and the denominator, $8$, must be a factor of the leading coefficient ($a_n$). Let's examine each function:

Function 1: $f(x)=24 x^7+3 x^6+4 x^3-x-28$

• The constant term ($a_0$) is $-28$.
• The leading coefficient ($a_n$) is $24$.

For $-\frac{7}{8}$ to be a potential rational root, $p=-7$ must divide $a_0=-28$, and $q=8$ must divide $a_n=24$.

• Does $-7$ divide $-28$? Yes, $-28 \div -7 = 4$. So, $-7$ is a factor of $-28$.
• Does $8$ divide $24$? Yes, $24 \div 8 = 3$. So, $8$ is a factor of $24$.

Since both conditions are met, $-\frac{7}{8}$ is a potential rational root for this function. This looks promising!

Function 2: $f(x)=28 x^7+3 x^6+4 x^3-x-24$

• The constant term ($a_0$) is $-24$.
• The leading coefficient ($a_n$) is $28$.

For $-\frac{7}{8}$ to be a potential rational root, $p=-7$ must divide $a_0=-24$, and $q=8$ must divide $a_n=28$.

• Does $-7$ divide $-24$? No, $-24$ is not divisible by $-7$ without a remainder. So, $-7$ is not a factor of $-24$.
• Does $8$ divide $28$? No, $28$ is not divisible by $8$ without a remainder. So, $8$ is not a factor of $28$.

Since at least one (in this case, both) of the conditions is not met, $-\frac{7}{8}$ is not a potential rational root for this function according to the Rational Root Theorem.

Function 3: $f(x)=30 x^7+3 x^6+4 x^3-x-56$

• The constant term ($a_0$) is $-56$.
• The leading coefficient ($a_n$) is $30$.

For $-\frac{7}{8}$ to be a potential rational root, $p=-7$ must divide $a_0=-56$, and $q=8$ must divide $a_n=30$.

• Does $-7$ divide $-56$? Yes, $-56 \div -7 = 8$. So, $-7$ is a factor of $-56$.
• Does $8$ divide $30$? No, $30$ is not divisible by $8$ without a remainder. So, $8$ is not a factor of $30$.

Since the second condition is not met, $-\frac{7}{8}$ is not a potential rational root for this function according to the Rational Root Theorem.

Conclusion: Identifying the Correct Function

After carefully applying the Rational Root Theorem to each function, we've found that only the first function satisfies the conditions for $-\frac{7}{8}$ to be a potential rational root. In the first function, $f(x)=24 x^7+3 x^6+4 x^3-x-28$, the numerator $-7$ is a factor of the constant term $-28$, and the denominator $8$ is a factor of the leading coefficient $24$. The other two functions failed to meet at least one of these criteria. Therefore, according to the Rational Root Theorem, $-\frac{7}{8}$ is a potential rational root of $f(x)=24 x^7+3 x^6+4 x^3-x-28$. This theorem is a fantastic shortcut for mathematicians, saving us tons of time and effort when exploring the roots of polynomial equations. It’s a cornerstone for understanding polynomial behavior and solving complex algebraic problems.

For further exploration into polynomial roots and the Rational Root Theorem, you can check out resources like ** Khan Academy's explanation of the Rational Root Theorem**.