Question
Which statements are true about the polynomial 4x3 – 6x2 + 8x – 12? Check all that apply.
The terms 4x3 and 8x have a common factor.
The terms 4x3 and – 6x2 have a common factor.
The polynomial is prime.
The factored polynomial is (2x2 – 3)(2x + 4).
The polynomial can be grouped in different ways to factor by grouping
The terms 4x3 and 8x have a common factor.
The terms 4x3 and – 6x2 have a common factor.
The polynomial is prime.
The factored polynomial is (2x2 – 3)(2x + 4).
The polynomial can be grouped in different ways to factor by grouping
Answers
Answered by
GPT-4o mini
To analyze the polynomial \(4x^3 - 6x^2 + 8x - 12\), let's evaluate each statement one by one:
1. **The terms \(4x^3\) and \(8x\) have a common factor.**
- **True.** The greatest common factor of \(4x^3\) and \(8x\) is \(4x\).
2. **The terms \(4x^3\) and \(-6x^2\) have a common factor.**
- **True.** The greatest common factor of \(4x^3\) and \(-6x^2\) is \(2x^2\).
3. **The polynomial is prime.**
- **False.** A polynomial is considered prime if it cannot be factored into the product of two non-constant polynomials. Since we can factor this polynomial, it is not prime.
4. **The factored polynomial is \((2x^2 - 3)(2x + 4)\).**
- **False.** The correct factorization involves looking for common factors and possibly grouping. The polynomial can indeed be factored, but the provided factorization is incorrect.
5. **The polynomial can be grouped in different ways to factor by grouping.**
- **True.** This polynomial can indeed be factored by grouping, and sometimes it can be rearranged to allow different groupings that are valid.
In conclusion, the true statements 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 polynomial can be grouped in different ways to factor by grouping.
1. **The terms \(4x^3\) and \(8x\) have a common factor.**
- **True.** The greatest common factor of \(4x^3\) and \(8x\) is \(4x\).
2. **The terms \(4x^3\) and \(-6x^2\) have a common factor.**
- **True.** The greatest common factor of \(4x^3\) and \(-6x^2\) is \(2x^2\).
3. **The polynomial is prime.**
- **False.** A polynomial is considered prime if it cannot be factored into the product of two non-constant polynomials. Since we can factor this polynomial, it is not prime.
4. **The factored polynomial is \((2x^2 - 3)(2x + 4)\).**
- **False.** The correct factorization involves looking for common factors and possibly grouping. The polynomial can indeed be factored, but the provided factorization is incorrect.
5. **The polynomial can be grouped in different ways to factor by grouping.**
- **True.** This polynomial can indeed be factored by grouping, and sometimes it can be rearranged to allow different groupings that are valid.
In conclusion, the true statements 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 polynomial can be grouped in different ways to factor by grouping.