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

In this article, we will delve into the world of polynomials and analyze the given statement about the polynomial $4x^2 - 6x^2 + 8x - 12$. We will examine each statement and determine its validity.

Understanding Polynomials

A polynomial is an expression consisting of variables and coefficients combined using only addition, subtraction, and multiplication. The variables in a polynomial are often represented by letters such as $x$ or $y$, and the coefficients are numbers that are multiplied with the variables.

The Given Polynomial

The given polynomial is $4x^2 - 6x^2 + 8x - 12$. This polynomial consists of four terms: $4x^2$, $-6x^2$, $8x$, and $-12$.

Statement 1: The terms $4x^2$ and $-6x^2$ have a common factor.

To determine if the terms $4x^2$ and $-6x^2$ have a common factor, we need to find the greatest common factor (GCF) of the coefficients $4$ and $-6$. The GCF of $4$ and $-6$ is $2$. However, the GCF of $2x^2$ is $2x^2$. Since the GCF of the coefficients is $2$, but the GCF of the terms is $2x^2$, the terms $4x^2$ and $-6x^2$ do not have a common factor.

Statement 2: The terms $4x^2$ and $8x$ have a common factor.

To determine if the terms $4x^2$ and $8x$ have a common factor, we need to find the greatest common factor (GCF) of the coefficients $4$ and $8$. The GCF of $4$ and $8$ is $4$. However, the GCF of $x^2$ and $x$ is $x$. Since the GCF of the coefficients is $4$, but the GCF of the terms is $x$, the terms $4x^2$ and $8x$ do not have a common factor.

Statement 3: The terms $-6x^2$ and $8x$ have a common factor.

To determine if the terms $-6x^2$ and $8x$ have a common factor, we need to find the greatest common factor (GCF) of the coefficients $-6$ and $8$. The GCF of $-6$ and $8$ is $2$. However, the GCF of $x^2$ and $x$ is $x$. Since the GCF of the coefficients is $2$, but the GCF of the terms is $x$, the terms $-6x^2$ and $8x$ do not have a common factor.

Conclusion

In conclusion, none of the given statements are true. The terms $4x^2$ and $-6x^2$ do not have a common factor, the terms $4x^2$ and $8x$ do not have a common factor, and the terms $-6x^2$ and $8x$ do not have a common factor.

Key Takeaways

• The greatest common factor (GCF) of the coefficients of a polynomial is not always the same as the GCF of the terms.
• To determine if two terms have a common factor, we need to find the GCF of the coefficients and the GCF of the terms.
• If the GCF of the coefficients is not the same as the GCF of the terms, then the terms do not have a common factor.

Final Thoughts

In this article, we will continue to explore the world of polynomials and answer some frequently asked questions about polynomial analysis.

Q: What is a polynomial?

A polynomial is an expression consisting of variables and coefficients combined using only addition, subtraction, and multiplication. The variables in a polynomial are often represented by letters such as $x$ or $y$, and the coefficients are numbers that are multiplied with the variables.

Q: What is the difference between a polynomial and an algebraic expression?

A polynomial is a specific type of algebraic expression that consists of variables and coefficients combined using only addition, subtraction, and multiplication. An algebraic expression, on the other hand, can include any combination of variables, coefficients, and mathematical operations.

Q: How do I determine the degree of a polynomial?

The degree of a polynomial is the highest power of the variable in the polynomial. For example, in the polynomial $x^2 + 3x + 2$, the degree is 2 because the highest power of $x$ is 2.

Q: What is the leading coefficient of a polynomial?

The leading coefficient of a polynomial is the coefficient of the term with the highest power of the variable. For example, in the polynomial $x^2 + 3x + 2$, the leading coefficient is 1 because it is the coefficient of the term with the highest power of $x$.

Q: How do I factor a polynomial?

Factoring a polynomial involves expressing it as a product of simpler polynomials. There are several methods for factoring polynomials, including:

• Factoring out the greatest common factor (GCF) of the terms
• Using the difference of squares formula
• Using the sum and difference of cubes formula
• Using synthetic division

Q: What is the difference between a monomial and a polynomial?

A monomial is a single term that consists of a variable and a coefficient. A polynomial, on the other hand, is an expression that consists of two or more terms. For example, $x^2$ is a monomial, while $x^2 + 3x + 2$ is a polynomial.

Q: How do I add and subtract polynomials?

To add and subtract polynomials, you need to combine like terms. Like terms are terms that have the same variable and exponent. For example, in the polynomials $x^2 + 3x + 2$ and $x^2 + 2x + 1$, the like terms are $x^2$ and $3x$ and $2x$.

Q: What is the zero polynomial?

The zero polynomial is a polynomial that has a degree of 0 and a coefficient of 0. It is often denoted as $0$ or $0x^0$. The zero polynomial is a special case because it is the only polynomial that is equal to 0 for all values of the variable.

Q: How do I multiply polynomials?

To multiply polynomials, you need to use the distributive property and multiply each term in one polynomial by each term in the other polynomial. For example, to multiply the polynomials $x^2 + 3x + 2$ and $x + 2$, you would multiply each term in the first polynomial by each term in the second polynomial.

Q: What is the difference between a polynomial and a rational expression?

A polynomial is an expression that consists of variables and coefficients combined using only addition, subtraction, and multiplication. A rational expression, on the other hand, is an expression that consists of a fraction of two polynomials. For example, $\frac{x^2 + 3x + 2}{x + 2}$ is a rational expression.

Conclusion

In this article, we answered some frequently asked questions about polynomial analysis. We discussed the definition of a polynomial, the difference between a polynomial and an algebraic expression, and how to determine the degree of a polynomial. We also covered how to factor a polynomial, add and subtract polynomials, and multiply polynomials. Finally, we discussed the difference between a polynomial and a rational expression.