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

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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 as $4x^3 - 6x^2 + 8x - 12$. To understand the properties of this polynomial, we need to analyze its terms.

Term Analysis

The polynomial has four terms: $4x^3$, $-6x^2$, $8x$, and $-12$. Each term has a coefficient and a variable part. The coefficients are 4, -6, 8, and -12, respectively. The variables are $x^3$, $x^2$, $x$, and the constant term.

Common Factors

The first statement claims that the terms $4x^3$ and $8x$ have a common factor. To determine if this is true, we need to find the greatest common factor (GCF) of these two terms.

The GCF of $4x^3$ and $8x$ is 4x. This is because 4 is the greatest common factor of the coefficients 4 and 8, and x is the greatest common factor of the variables $x^3$ and x.

Therefore, the statement that the terms $4x^3$ and $8x$ have a common factor is true.

Common Factors Between Terms

The second statement claims that the terms $4x^3$ and $-6x^2$ have a common factor. To determine if this is true, we need to find the GCF of these two terms.

The GCF of $4x^3$ and $-6x^2$ is 2x^2. This is because 2 is the greatest common factor of the coefficients 4 and -6, and x^2 is the greatest common factor of the variables $x^3$ and $x^2$.

Therefore, the statement that the terms $4x^3$ and $-6x^2$ have a common factor is true.

Common Factors with the Constant Term

The third statement claims that the terms $4x^3$ and $-12$ have a common factor. To determine if this is true, we need to find the GCF of these two terms.

The GCF of $4x^3$ and $-12$ is 4. This is because 4 is the greatest common factor of the coefficients 4 and -12.

Therefore, the statement that the terms $4x^3$ and $-12$ have a common factor is true.

Common Factors with the Other Terms

The fourth statement claims that the terms $-6x^2$ and $8x$ have a common factor. To determine if this is true, we need to find the GCF of these two terms.

The GCF of $-6x^2$ and $8x$ is 2x. This is because 2 is the greatest common factor of the coefficients -6 and 8, and x is the greatest common factor of the variables $x^2$ and x.

Therefore, the statement that the terms $-6x^2$ and $8x$ have a common factor is true.

Common Factors with the Constant Term

The fifth statement claims that the terms $-6x^2$ and $-12$ have a common factor. To determine if this is true, we need to find the GCF of these two terms.

The GCF of $-6x^2$ and $-12$ is -6. This is because -6 is the greatest common factor of the coefficients -6 and -12.

Therefore, the statement that the terms $-6x^2$ and $-12$ have a common factor is true.

Common Factors with the Other Terms

The sixth statement claims that the terms $8x$ and $-12$ have a common factor. To determine if this is true, we need to find the GCF of these two terms.

The GCF of $8x$ and $-12$ is 4. This is because 4 is the greatest common factor of the coefficients 8 and -12.

Therefore, the statement that the terms $8x$ and $-12$ have a common factor is true.

Conclusion

In conclusion, the statements that the terms $4x^3$ and $8x$ have a common factor, the terms $4x^3$ and $-6x^2$ have a common factor, the terms $4x^3$ and $-12$ have a common factor, the terms $-6x^2$ and $8x$ have a common factor, the terms $-6x^2$ and $-12$ have a common factor, and the terms $8x$ and $-12$ have a common factor are all true.

Key Takeaways

  • The terms $4x^3$ and $8x$ have a common factor of 4x.
  • The terms $4x^3$ and $-6x^2$ have a common factor of 2x^2.
  • The terms $4x^3$ and $-12$ have a common factor of 4.
  • The terms $-6x^2$ and $8x$ have a common factor of 2x.
  • The terms $-6x^2$ and $-12$ have a common factor of -6.
  • The terms $8x$ and $-12$ have a common factor of 4.

Final Thoughts

In this article, we will address some frequently asked questions (FAQs) about the polynomial $4x^3 - 6x^2 + 8x - 12$.

Q: What is the degree of the polynomial?

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

Q: What are the coefficients of the polynomial?

A: The coefficients of the polynomial are 4, -6, 8, and -12.

Q: What are the variables of the polynomial?

A: The variables of the polynomial are $x^3$, $x^2$, $x$, and the constant term.

Q: What is the greatest common factor (GCF) of the terms $4x^3$ and $8x$?

A: The GCF of the terms $4x^3$ and $8x$ is 4x.

Q: What is the greatest common factor (GCF) of the terms $4x^3$ and $-6x^2$?

A: The GCF of the terms $4x^3$ and $-6x^2$ is 2x^2.

Q: What is the greatest common factor (GCF) of the terms $4x^3$ and $-12$?

A: The GCF of the terms $4x^3$ and $-12$ is 4.

Q: What is the greatest common factor (GCF) of the terms $-6x^2$ and $8x$?

A: The GCF of the terms $-6x^2$ and $8x$ is 2x.

Q: What is the greatest common factor (GCF) of the terms $-6x^2$ and $-12$?

A: The GCF of the terms $-6x^2$ and $-12$ is -6.

Q: What is the greatest common factor (GCF) of the terms $8x$ and $-12$?

A: The GCF of the terms $8x$ and $-12$ is 4.

Q: Can you factor the polynomial?

A: Yes, the polynomial can be factored as follows:

4x3βˆ’6x2+8xβˆ’12=2x(2x2βˆ’3x+4)4x^3 - 6x^2 + 8x - 12 = 2x(2x^2 - 3x + 4)

Q: What is the value of the polynomial when x = 1?

A: To find the value of the polynomial when x = 1, we substitute x = 1 into the polynomial:

4(1)3βˆ’6(1)2+8(1)βˆ’12=4βˆ’6+8βˆ’12=βˆ’64(1)^3 - 6(1)^2 + 8(1) - 12 = 4 - 6 + 8 - 12 = -6

Therefore, the value of the polynomial when x = 1 is -6.

Q: What is the value of the polynomial when x = 2?

A: To find the value of the polynomial when x = 2, we substitute x = 2 into the polynomial:

4(2)3βˆ’6(2)2+8(2)βˆ’12=32βˆ’24+16βˆ’12=124(2)^3 - 6(2)^2 + 8(2) - 12 = 32 - 24 + 16 - 12 = 12

Therefore, the value of the polynomial when x = 2 is 12.

Q: Can you graph the polynomial?

A: Yes, the polynomial can be graphed using a graphing calculator or a computer algebra system. The graph of the polynomial is a cubic curve that opens upward.

Q: What are some real-world applications of the polynomial?

A: The polynomial has many real-world applications, including:

  • Modeling population growth
  • Modeling the motion of objects
  • Modeling financial data
  • Modeling scientific data

These are just a few examples of the many real-world applications of the polynomial.