Which Statements Are True About The Polynomial $4x^3 - 6x^2 + 8x - 12$? Check All That Apply.- The Terms $4x^3$ And \$8x$[/tex\] Have A Common Factor.- The Terms $4x^3$ And $-6x^2$ Have A Common

Feb 27, 2025 by ADMIN 204 views

**Exploring the Properties of a Given Polynomial**

In this article, we will delve into the properties of a given polynomial, specifically the polynomial $4x^3 - 6x^2 + 8x - 12$. We will examine the statements provided and determine which ones are true.

Understanding the Polynomial

The given polynomial is a cubic polynomial, meaning it has a degree of 3. It is expressed in the form $ax^3 + bx^2 + cx + d$, where $a = 4$, $b = -6$, $c = 8$, and $d = -12$.

Statement 1: The terms $4x^3$ and $8x$ have a common factor.

To determine if this statement is true, we need to examine the factors of the terms $4x^3$ and $8x$. The term $4x^3$ can be factored as $4x^2(x)$, and the term $8x$ can be factored as $8(x)$. We can see that both terms have a common factor of $4x^2$, but not $x$. However, we can also factor out a common factor of $4$ from both terms, resulting in $4x^3 = 4x(x^2)$ and $8x = 4(2x)$. Therefore, the statement that the terms $4x^3$ and $8x$ have a common factor is true.

Statement 2: The terms $4x^3$ and $-6x^2$ have a common factor.

To determine if this statement is true, we need to examine the factors of the terms $4x^3$ and $-6x^2$. The term $4x^3$ can be factored as $4x^2(x)$, and the term $-6x^2$ can be factored as $-6(x^2)$. We can see that both terms have a common factor of $x^2$, but not $x$. Therefore, the statement that the terms $4x^3$ and $-6x^2$ have a common factor is true.

Statement 3: The polynomial can be factored as $(x-1)(4x^2+8x-12)$.

To determine if this statement is true, we need to examine the factors of the polynomial. We can start by factoring out a common factor of $4$ from the polynomial, resulting in $4(x^3 - \frac{3}{2}x^2 + 2x - 3)$. We can then try to factor the remaining polynomial, but it does not factor easily. Therefore, the statement that the polynomial can be factored as $(x-1)(4x^2+8x-12)$ is false.

Conclusion

In conclusion, we have examined the properties of the given polynomial and determined which statements are true. The statements that the terms $4x^3$ and $8x$ have a common factor, and the terms $4x^3$ and $-6x^2$ have a common factor, are both true. However, the statement that the polynomial can be factored as $(x-1)(4x^2+8x-12)$ is false.

Additional Analysis

To further analyze the polynomial, we can try to factor it using different methods. One method is to use the Rational Root Theorem, which states that any rational root of a polynomial must be of the form $\frac{p}{q}$, where $p$ is a factor of the constant term and $q$ is a factor of the leading coefficient. In this case, the constant term is $-12$, and the leading coefficient is $4$. Therefore, any rational root of the polynomial must be of the form $\frac{p}{q}$, where $p$ is a factor of $-12$ and $q$ is a factor of $4$.

Factoring the Polynomial

Using the Rational Root Theorem, we can try to find the rational roots of the polynomial. We can start by listing the possible rational roots, which are $\pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 12$. We can then try to substitute each of these values into the polynomial to see if it is a root. After trying each of the possible rational roots, we find that $x = 1$ is a root of the polynomial.

Factoring the Polynomial (continued)

Now that we have found one root of the polynomial, we can try to factor the polynomial using polynomial long division. We can divide the polynomial by $(x-1)$ to get the quotient $4x^2 + 8x - 12$. We can then try to factor the quotient, but it does not factor easily. Therefore, the polynomial cannot be factored into the product of two binomials.

Conclusion (continued)

In our previous article, we examined the properties of a given polynomial and determined which statements are true. In this article, we will continue to explore the properties of the polynomial and answer some frequently asked questions.

Q: What is the degree of the polynomial?

A: The degree of the polynomial is 3, which means it is a cubic polynomial.

Q: Can the polynomial be factored into the product of two binomials?

A: No, the polynomial cannot be factored into the product of two binomials. We tried to factor the polynomial using different methods, but it does not factor easily.

Q: What are the factors of the polynomial?

A: The factors of the polynomial are $4x^3 - 6x^2 + 8x - 12$, but it does not factor easily.

Q: Can the polynomial be factored using the Rational Root Theorem?

A: Yes, the polynomial can be factored using the Rational Root Theorem. We found that $x = 1$ is a root of the polynomial.

Q: How can we factor the polynomial using polynomial long division?

A: We can factor the polynomial using polynomial long division by dividing the polynomial by $(x-1)$ to get the quotient $4x^2 + 8x - 12$.

Q: Can the quotient be factored further?

A: No, the quotient $4x^2 + 8x - 12$ does not factor easily.

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

A: The possible rational roots of the polynomial are $\pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 12$.

Q: How can we determine if a number is a root of the polynomial?

A: We can determine if a number is a root of the polynomial by substituting the number into the polynomial and checking if the result is equal to zero.

Q: What is the significance of the Rational Root Theorem?

A: The Rational Root Theorem is a useful tool for finding the roots of a polynomial. It states that any rational root of a polynomial must be of the form $\frac{p}{q}$, where $p$ is a factor of the constant term and $q$ is a factor of the leading coefficient.

Q: Can the polynomial be used to model real-world situations?

A: Yes, the polynomial can be used to model real-world situations. For example, it can be used to model the growth of a population or the cost of a product.

Conclusion

In conclusion, we have explored the properties of a given polynomial and answered some frequently asked questions. We have determined that the polynomial cannot be factored into the product of two binomials, but it can be factored using the Rational Root Theorem. We have also discussed the significance of the Rational Root Theorem and how it can be used to find the roots of a polynomial.

Additional Resources

For more information on polynomials and the Rational Root Theorem, please see the following resources:

Practice Problems

Try the following practice problems to test your understanding of the material:

Factor the polynomial $x^2 + 5x + 6$.
Find the roots of the polynomial $x^2 + 4x + 4$.
Use the Rational Root Theorem to find the roots of the polynomial $x^3 - 2x^2 - 5x + 6$.