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]
