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

Understanding the Polynomial

The given polynomial is $4x^3 - 6x^2 + 8x - 12$. To determine which statements are true about this polynomial, we need to carefully examine each statement and use our knowledge of factoring polynomials.

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

To check if the terms $4x^3$ and $8x$ have a common factor, we need to look for the greatest common factor (GCF) of the coefficients and the lowest power of $x$ that divides both terms. The coefficients of the two terms are 4 and 8, respectively. The GCF of 4 and 8 is 4. The lowest power of $x$ that divides both terms is $x$. Therefore, the common factor of the terms $4x^3$ and $8x$ is $4x$. This means that statement 1 is true.

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

To check if the terms $4x^3$ and $-6x^2$ have a common factor, we need to look for the greatest common factor (GCF) of the coefficients and the lowest power of $x$ that divides both terms. The coefficients of the two terms are 4 and -6, respectively. The GCF of 4 and -6 is 2. The lowest power of $x$ that divides both terms is $x^2$. Therefore, the common factor of the terms $4x^3$ and $-6x^2$ is $2x^2$. This means that statement 2 is true.

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

This statement is a rewording of statement 2. Since we have already determined that statement 2 is true, this statement is also true.

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

This statement is a rewording of statement 1. Since we have already determined that statement 1 is true, this statement is also true.

Conclusion

In conclusion, the statements that are true about the polynomial $4x^3 - 6x^2 + 8x - 12$ are:

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

These statements are all true because the polynomial can be factored by grouping, and the common factors of the terms can be identified.

Factoring the Polynomial by Grouping

To factor the polynomial by grouping, we can group the first two terms and the last two terms:

$4x^3 - 6x^2 + 8x - 12$

$= (4x^3 - 6x^2) + (8x - 12)$

$= 2x^2(2x - 3) + 4(2x - 3)$

$= (2x^2 + 4)(2x - 3)$

$= 2(x^2 + 2)(2x - 3)$

$= 2(x^2 + 2)(x - \frac{3}{2})$

This is the factored form of the polynomial.

Conclusion

In conclusion, the polynomial $4x^3 - 6x^2 + 8x - 12$ can be factored by grouping, and the common factors of the terms can be identified. The statements that are true about the polynomial are:

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

These statements are all true because the polynomial can be factored by grouping, and the common factors of the terms can be identified.

Understanding the Polynomial

The given polynomial is $4x^3 - 6x^2 + 8x - 12$. To determine which statements are true about this polynomial, we need to carefully examine each statement and use our knowledge of factoring polynomials.

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

To check if the terms $4x^3$ and $8x$ have a common factor, we need to look for the greatest common factor (GCF) of the coefficients and the lowest power of $x$ that divides both terms. The coefficients of the two terms are 4 and 8, respectively. The GCF of 4 and 8 is 4. The lowest power of $x$ that divides both terms is $x$. Therefore, the common factor of the terms $4x^3$ and $8x$ is $4x$. This means that statement 1 is true.

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

To check if the terms $4x^3$ and $-6x^2$ have a common factor, we need to look for the greatest common factor (GCF) of the coefficients and the lowest power of $x$ that divides both terms. The coefficients of the two terms are 4 and -6, respectively. The GCF of 4 and -6 is 2. The lowest power of $x$ that divides both terms is $x^2$. Therefore, the common factor of the terms $4x^3$ and $-6x^2$ is $2x^2$. This means that statement 2 is true.

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

This statement is a rewording of statement 2. Since we have already determined that statement 2 is true, this statement is also true.

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

This statement is a rewording of statement 1. Since we have already determined that statement 1 is true, this statement is also true.

Conclusion

In conclusion, the statements that are true about the polynomial $4x^3 - 6x^2 + 8x - 12$ are:

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

These statements are all true because the polynomial can be factored by grouping, and the common factors of the terms can be identified.

Factoring the Polynomial by Grouping

To factor the polynomial by grouping, we can group the first two terms and the last two terms:

$4x^3 - 6x^2 + 8x - 12$

$= (4x^3 - 6x^2) + (8x - 12)$

$= 2x^2(2x - 3) + 4(2x - 3)$

$= (2x^2 + 4)(2x - 3)$

$= 2(x^2 + 2)(2x - 3)$

$= 2(x^2 + 2)(x - \frac{3}{2})$

This is the factored form of the polynomial.

Q&A

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

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

Q: What is the lowest power of $x$ that divides both terms $4x^3$ and $8x$?

A: The lowest power of $x$ that divides both terms $4x^3$ and $8x$ is $x$.

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

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

Q: How can we factor the polynomial $4x^3 - 6x^2 + 8x - 12$?

A: We can factor the polynomial by grouping. We can group the first two terms and the last two terms, and then factor out the common factors.

Q: What is the factored form of the polynomial $4x^3 - 6x^2 + 8x - 12$?

A: The factored form of the polynomial $4x^3 - 6x^2 + 8x - 12$ is $2(x^2 + 2)(x - \frac{3}{2})$.

Q: Why is it important to identify the common factors of the terms in a polynomial?

A: It is important to identify the common factors of the terms in a polynomial because it allows us to factor the polynomial and simplify it.

Q: Can we always factor a polynomial by grouping?

A: No, we cannot always factor a polynomial by grouping. However, it is a useful technique to try when factoring a polynomial.

Q: What are some other techniques for factoring polynomials?

A: Some other techniques for factoring polynomials include factoring by difference of squares, factoring by sum and difference of cubes, and factoring by grouping.

Q: How can we determine which technique to use when factoring a polynomial?

A: We can determine which technique to use by examining the polynomial and looking for common factors or patterns that can be used to factor it.

Q: Can we always find the factored form of a polynomial?

A: No, we cannot always find the factored form of a polynomial. However, we can often find a simplified form of the polynomial that is easier to work with.

Q: Why is it important to understand how to factor polynomials?

A: It is important to understand how to factor polynomials because it allows us to simplify complex expressions and solve equations more easily.

Q: Can you give an example of a polynomial that cannot be factored by grouping?

A: Yes, the polynomial $x^2 + 2x + 1$ cannot be factored by grouping. However, it can be factored as $(x + 1)^2$.

Q: Can you give an example of a polynomial that can be factored by grouping?

A: Yes, the polynomial $x^3 + 2x^2 + 3x + 6$ can be factored by grouping. We can group the first two terms and the last two terms, and then factor out the common factors.

Q: How can we use factoring to solve equations?

A: We can use factoring to solve equations by setting the factored form of the polynomial equal to zero and solving for the variable.

Q: Can you give an example of an equation that can be solved using factoring?

A: Yes, the equation $x^2 + 4x + 4 = 0$ can be solved using factoring. We can factor the left-hand side of the equation as $(x + 2)^2 = 0$, and then solve for $x$.

Q: Why is it important to understand how to solve equations using factoring?

A: It is important to understand how to solve equations using factoring because it allows us to find the solutions to equations more easily and efficiently.

Q: Can you give an example of a polynomial that can be factored using the difference of squares formula?

A: Yes, the polynomial $x^2 - 4$ can be factored using the difference of squares formula. We can write it as $(x - 2)(x + 2)$.

Q: Can you give an example of a polynomial that can be factored using the sum and difference of cubes formula?

A: Yes, the polynomial $x^3 + 8$ can be factored using the sum and difference of cubes formula. We can write it as $(x + 2)(x^2 - 2x + 4)$.

Q: Why is it important to understand how to factor polynomials using different techniques?

A: It is important to understand how to factor polynomials using different techniques because it allows us to simplify complex expressions and solve equations more easily and efficiently.

Q: Can you give an example of a polynomial that can be factored using the rational root theorem?

A: Yes, the polynomial $x^3 - 6x^2 + 9x - 6$ can be factored using the rational root theorem. We can find the rational roots of the polynomial and then factor it.

Q: Why is it important to understand how to factor polynomials using the rational root theorem?

A: It is important to understand