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Find the HCF (Highest Common Factor) of the following pairs of polynomials:

(a) [tex]12(x^3 + x^2 + x + 1)[/tex] and [tex]18(x^4 - 1)[/tex]

(c) [tex]4(x^4 - 1)[/tex] and [tex]6(x^3 - x^2 - x + 1)[/tex]

(e) [tex]18(6x^4 + x^3 - x^2)[/tex] and [tex]45(2x^6 + 3x^5 + x^4)[/tex]

(g) [tex]2x^2 - x - 1[/tex] and [tex]4x^2 + 8x + 3[/tex]

(i) [tex]2x^2 - 18[/tex] and [tex]x^2 - 2x - 3[/tex]

Answer :

To find the Highest Common Factor (HCF) of pairs of polynomials, let's look at each pair separately and understand what their HCF is.

### Pair (a)

Polynomials:
- [tex]\(12(x^3 + x^2 + x + 1)\)[/tex]
- [tex]\(18(x^4 - 1)\)[/tex]

HCF:
- The highest common factor for these two polynomials is:
[tex]\[
6(x^3 + x^2 + x + 1)
\][/tex]

### Pair (c)

Polynomials:
- [tex]\(4(x^4 - 1)\)[/tex]
- [tex]\(6(x^3 - x^2 - x + 1)\)[/tex]

HCF:
- The highest common factor for these two polynomials is:
[tex]\[
2(x^2 - 1)
\][/tex]

### Pair (e)

Polynomials:
- [tex]\(18(6x^4 + x^3 - x^2)\)[/tex]
- [tex]\(45(2x^6 + 3x^5 + x^4)\)[/tex]

HCF:
- The highest common factor for these two polynomials is:
[tex]\[
18x^3 + 9x^2
\][/tex]

### Pair (g)

Polynomials:
- [tex]\(2x^2 - x - 1\)[/tex]
- [tex]\(4x^2 + 8x + 3\)[/tex]

HCF:
- The highest common factor for these two polynomials is:
[tex]\[
2x + 1
\][/tex]

### Pair (i)

Polynomials:
- [tex]\(2x^2 - 18\)[/tex]
- [tex]\(x^2 - 2x - 3\)[/tex]

HCF:
- The highest common factor for these two polynomials is:
[tex]\[
x - 3
\][/tex]

These results reflect the common factors of each set of polynomials, showing how they each factor down to the expressions given as the HCF in each case.

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