According To The Rational Root Theorem, Which Is A Factor Of The Polynomial $f(x)=3x^3-5x^2-12x+20$?A. $2x+1$B. $2x-1$C. $3x+5$D. $3x-5$

According to the Rational Root Theorem, Which is a Factor of the Polynomial $f(x)=3x^3-5x^2-12x+20$?

The Rational Root Theorem is a fundamental concept in algebra that helps us determine the possible rational roots of a polynomial equation. This theorem is a powerful tool for solving polynomial equations and is widely used in various fields of mathematics and science. In this article, we will apply the Rational Root Theorem to the polynomial $f(x)=3x^3-5x^2-12x+20$ and determine which of the given options is a factor of this polynomial.

Understanding the Rational Root Theorem

The Rational Root Theorem states that if a rational number $p/q$ is a root of the polynomial $f(x)$, where $p$ and $q$ are integers and $q$ is non-zero, then $p$ must be a factor of the constant term of $f(x)$, and $q$ must be a factor of the leading coefficient of $f(x)$. In other words, the rational root of a polynomial must be of the form $p/q$, where $p$ is a factor of the constant term and $q$ is a factor of the leading coefficient.

Applying the Rational Root Theorem to the Polynomial $f(x)=3x^3-5x^2-12x+20$

To apply the Rational Root Theorem to the polynomial $f(x)=3x^3-5x^2-12x+20$, we need to find the factors of the constant term and the leading coefficient. The constant term of $f(x)$ is 20, and its factors are $\pm 1, \pm 2, \pm 4, \pm 5, \pm 10, \pm 20$. The leading coefficient of $f(x)$ is 3, and its factors are $\pm 1, \pm 3$.

Possible Rational Roots

Using the Rational Root Theorem, we can determine the possible rational roots of the polynomial $f(x)=3x^3-5x^2-12x+20$. The possible rational roots are of the form $p/q$, where $p$ is a factor of 20 and $q$ is a factor of 3. Therefore, the possible rational roots are:

• $\pm 1/1$
• $\pm 2/1$
• $\pm 4/1$
• $\pm 5/1$
• $\pm 10/1$
• $\pm 20/1$
• $\pm 1/3$
• $\pm 2/3$
• $\pm 4/3$
• $\pm 5/3$
• $\pm 10/3$
• $\pm 20/3$

Testing the Possible Rational Roots

To determine which of the possible rational roots is actually a root of the polynomial $f(x)=3x^3-5x^2-12x+20$, we need to test each of them by substituting them into the polynomial and checking if the result is zero. We can use synthetic division or direct substitution to test each possible rational root.

Testing Option A: $2x+1$

Let's test option A: $2x+1$. We can substitute $x=-1/2$ into the polynomial $f(x)=3x^3-5x^2-12x+20$ and check if the result is zero.

$f(-1/2) = 3(-1/2)^3 - 5(-1/2)^2 - 12(-1/2) + 20$ $f(-1/2) = 3(-1/8) - 5(1/4) + 6 + 20$ $f(-1/2) = -3/8 - 5/4 + 6 + 20$ $f(-1/2) = -3/8 - 10/8 + 48/8 + 160/8$ $f(-1/2) = 195/8$

Since $f(-1/2) \neq 0$, option A: $2x+1$ is not a factor of the polynomial $f(x)=3x^3-5x^2-12x+20$.

Testing Option B: $2x-1$

Let's test option B: $2x-1$. We can substitute $x=1/2$ into the polynomial $f(x)=3x^3-5x^2-12x+20$ and check if the result is zero.

$f(1/2) = 3(1/2)^3 - 5(1/2)^2 - 12(1/2) + 20$ $f(1/2) = 3(1/8) - 5(1/4) - 6 + 20$ $f(1/2) = 3/8 - 5/4 - 6 + 20$ $f(1/2) = 3/8 - 10/8 - 48/8 + 160/8$ $f(1/2) = 103/8$

Since $f(1/2) \neq 0$, option B: $2x-1$ is not a factor of the polynomial $f(x)=3x^3-5x^2-12x+20$.

Testing Option C: $3x+5$

Let's test option C: $3x+5$. We can substitute $x=-5/3$ into the polynomial $f(x)=3x^3-5x^2-12x+20$ and check if the result is zero.

$f(-5/3) = 3(-5/3)^3 - 5(-5/3)^2 - 12(-5/3) + 20$ $f(-5/3) = 3(-125/27) - 5(25/9) + 20 + 20$ $f(-5/3) = -125/9 - 125/9 + 40 + 40$ $f(-5/3) = -250/9 + 80$ $f(-5/3) = -250/9 + 720/9$ $f(-5/3) = 470/9$

Since $f(-5/3) \neq 0$, option C: $3x+5$ is not a factor of the polynomial $f(x)=3x^3-5x^2-12x+20$.

Testing Option D: $3x-5$

Let's test option D: $3x-5$. We can substitute $x=5/3$ into the polynomial $f(x)=3x^3-5x^2-12x+20$ and check if the result is zero.

$f(5/3) = 3(5/3)^3 - 5(5/3)^2 - 12(5/3) + 20$ $f(5/3) = 3(125/27) - 5(25/9) - 20 + 20$ $f(5/3) = 125/9 - 125/9 - 20 + 20$ $f(5/3) = 0$

Since $f(5/3) = 0$, option D: $3x-5$ is a factor of the polynomial $f(x)=3x^3-5x^2-12x+20$.

In conclusion, we have applied the Rational Root Theorem to the polynomial $f(x)=3x^3-5x^2-12x+20$ and determined that option D: $3x-5$ is a factor of this polynomial. We have tested each of the possible rational roots and found that only option D: $3x-5$ satisfies the condition of being a factor of the polynomial. Therefore, the correct answer is option D: $3x-5$.
Q&A: Rational Root Theorem and Polynomial Factorization

In our previous article, we applied the Rational Root Theorem to the polynomial $f(x)=3x^3-5x^2-12x+20$ and determined that option D: $3x-5$ is a factor of this polynomial. In this article, we will answer some frequently asked questions about the Rational Root Theorem and polynomial factorization.

Q: What is the Rational Root Theorem?

A: The Rational Root Theorem is a fundamental concept in algebra that helps us determine the possible rational roots of a polynomial equation. It states that if a rational number $p/q$ is a root of the polynomial $f(x)$, where $p$ and $q$ are integers and $q$ is non-zero, then $p$ must be a factor of the constant term of $f(x)$, and $q$ must be a factor of the leading coefficient of $f(x)$.

Q: How do I apply the Rational Root Theorem to a polynomial?

A: To apply the Rational Root Theorem to a polynomial, you need to find the factors of the constant term and the leading coefficient. Then, you can determine the possible rational roots by taking the ratio of the factors of the constant term and the leading coefficient.

Q: What are the possible rational roots of a polynomial?

A: The possible rational roots of a polynomial are of the form $p/q$, where $p$ is a factor of the constant term and $q$ is a factor of the leading coefficient.

Q: How do I test the possible rational roots of a polynomial?

A: To test the possible rational roots of a polynomial, you can substitute each possible root into the polynomial and check if the result is zero. You can use synthetic division or direct substitution to test each possible root.

Q: What is synthetic division?

A: Synthetic division is a method of dividing a polynomial by a linear factor. It is a shortcut for long division and is used to find the quotient and remainder of a polynomial division.

Q: How do I use synthetic division to test a possible rational root?

A: To use synthetic division to test a possible rational root, you need to write the coefficients of the polynomial in a row, followed by the possible root. Then, you multiply the first coefficient by the possible root and add the result to the second coefficient. You continue this process until you reach the last coefficient. If the result is zero, then the possible root is a root of the polynomial.

Q: What is the difference between a root and a factor of a polynomial?

A: A root of a polynomial is a value of $x$ that makes the polynomial equal to zero. A factor of a polynomial is a polynomial that divides the original polynomial without leaving a remainder.

Q: Can a polynomial have more than one factor?

A: Yes, a polynomial can have more than one factor. In fact, a polynomial can have multiple factors, each of which is a polynomial that divides the original polynomial without leaving a remainder.

Q: How do I find the factors of a polynomial?

A: To find the factors of a polynomial, you can use the Rational Root Theorem to determine the possible rational roots. Then, you can test each possible root using synthetic division or direct substitution to find the actual roots. Once you have found the roots, you can use polynomial division to find the factors.

Q: What is polynomial division?

A: Polynomial division is a method of dividing a polynomial by another polynomial. It is used to find the quotient and remainder of a polynomial division.

Q: How do I use polynomial division to find the factors of a polynomial?

A: To use polynomial division to find the factors of a polynomial, you need to divide the polynomial by each possible factor. If the result is zero, then the possible factor is a factor of the polynomial.

In conclusion, the Rational Root Theorem is a powerful tool for determining the possible rational roots of a polynomial equation. By applying the theorem and using synthetic division or direct substitution, you can find the actual roots of a polynomial and determine its factors. We hope that this Q&A article has helped you understand the Rational Root Theorem and polynomial factorization.