IDNLearn.com: Your reliable source for finding expert answers. Get the information you need from our community of experts who provide accurate and comprehensive answers to all your questions.
Sagot :
To find the Highest Common Factor (HCF) of the given pairs of polynomials, we need to determine the greatest common divisor of each pair. Below is a step-by-step explanation for each pair of polynomials:
1. For the pair: \(12(x^3 + x^2 + x + 1)\) and \(18(x^4 - 1)\)
Let's start with the polynomials:
[tex]\[ 12(x^3 + x^2 + x + 1) \quad \text{and} \quad 18(x^4 - 1) \][/tex]
The HCF of these polynomials is:
[tex]\[ 6(x^3 + x^2 + x + 1) \][/tex]
2. For the pair: \(4(x^4 - 1)\) and \(6(x^3 - x^2 - x + 1)\)
Let's start with the polynomials:
[tex]\[ 4(x^4 - 1) \quad \text{and} \quad 6(x^3 - x^2 - x + 1) \][/tex]
The HCF of these polynomials is:
[tex]\[ 2(x^2 - 1) \][/tex]
3. For the pair: \(18(6x^4 + x^3 - x^2)\) and \(45(2x^6 + 3x^5 + x^4)\)
Let's start with the polynomials:
[tex]\[ 18(6x^4 + x^3 - x^2) \quad \text{and} \quad 45(2x^6 + 3x^5 + x^4) \][/tex]
The HCF of these polynomials is:
[tex]\[ 18(x^3 + \frac{1}{2}x^2) = 18x^3 + 9x^2 \][/tex]
4. For the pair: \(2x^2 - x - 1\) and \(4x^2 + 8x + 3\)
Let's start with the polynomials directly:
[tex]\[ 2x^2 - x - 1 \quad \text{and} \quad 4x^2 + 8x + 3 \][/tex]
The HCF of these polynomials is:
[tex]\[ 2x + 1 \][/tex]
5. For the pair: \(2x^2 - 18\) and \(x^2 - 2x - 3\)
Let's start with the polynomials directly:
[tex]\[ 2x^2 - 18 \quad \text{and} \quad x^2 - 2x - 3 \][/tex]
The HCF of these polynomials is:
[tex]\[ x - 3 \][/tex]
Therefore, the HCFs for the given pairs of polynomials are as follows:
1. \(6x^3 + 6x^2 + 6x + 6\)
2. \(2x^2 - 2\)
3. \(18x^3 + 9x^2\)
4. \(2x + 1\)
5. [tex]\(x - 3\)[/tex]
1. For the pair: \(12(x^3 + x^2 + x + 1)\) and \(18(x^4 - 1)\)
Let's start with the polynomials:
[tex]\[ 12(x^3 + x^2 + x + 1) \quad \text{and} \quad 18(x^4 - 1) \][/tex]
The HCF of these polynomials is:
[tex]\[ 6(x^3 + x^2 + x + 1) \][/tex]
2. For the pair: \(4(x^4 - 1)\) and \(6(x^3 - x^2 - x + 1)\)
Let's start with the polynomials:
[tex]\[ 4(x^4 - 1) \quad \text{and} \quad 6(x^3 - x^2 - x + 1) \][/tex]
The HCF of these polynomials is:
[tex]\[ 2(x^2 - 1) \][/tex]
3. For the pair: \(18(6x^4 + x^3 - x^2)\) and \(45(2x^6 + 3x^5 + x^4)\)
Let's start with the polynomials:
[tex]\[ 18(6x^4 + x^3 - x^2) \quad \text{and} \quad 45(2x^6 + 3x^5 + x^4) \][/tex]
The HCF of these polynomials is:
[tex]\[ 18(x^3 + \frac{1}{2}x^2) = 18x^3 + 9x^2 \][/tex]
4. For the pair: \(2x^2 - x - 1\) and \(4x^2 + 8x + 3\)
Let's start with the polynomials directly:
[tex]\[ 2x^2 - x - 1 \quad \text{and} \quad 4x^2 + 8x + 3 \][/tex]
The HCF of these polynomials is:
[tex]\[ 2x + 1 \][/tex]
5. For the pair: \(2x^2 - 18\) and \(x^2 - 2x - 3\)
Let's start with the polynomials directly:
[tex]\[ 2x^2 - 18 \quad \text{and} \quad x^2 - 2x - 3 \][/tex]
The HCF of these polynomials is:
[tex]\[ x - 3 \][/tex]
Therefore, the HCFs for the given pairs of polynomials are as follows:
1. \(6x^3 + 6x^2 + 6x + 6\)
2. \(2x^2 - 2\)
3. \(18x^3 + 9x^2\)
4. \(2x + 1\)
5. [tex]\(x - 3\)[/tex]
Thank you for participating in our discussion. We value every contribution. Keep sharing knowledge and helping others find the answers they need. Let's create a dynamic and informative learning environment together. Your questions find answers at IDNLearn.com. Thanks for visiting, and come back for more accurate and reliable solutions.