Answer :
Let's find the Highest Common Factor (H.C.F.) for each pair of expressions step-by-step:
### Part (a) [tex]\(x^3 - 4x\)[/tex] and [tex]\(x^2 + 7x + 10\)[/tex]
1. The expressions are [tex]\(x^3 - 4x\)[/tex] and [tex]\(x^2 + 7x + 10\)[/tex].
2. Factorize both expressions:
- [tex]\(x^3 - 4x = x(x^2 - 4) = x(x - 2)(x + 2)\)[/tex]
- [tex]\(x^2 + 7x + 10 = (x + 2)(x + 5)\)[/tex]
3. Identify the common factors:
- The common factor in both expressions is [tex]\((x + 2)\)[/tex].
Thus, the H.C.F. of [tex]\(x^3 - 4x\)[/tex] and [tex]\(x^2 + 7x + 10\)[/tex] is [tex]\(\boxed{x + 2}\)[/tex].
### Part (b) [tex]\(a^2 - 6a + 6b - b^2\)[/tex] and [tex]\(b^2 + ab - 6b\)[/tex]
1. The expressions are [tex]\(a^2 - 6a + 6b - b^2\)[/tex] and [tex]\(b^2 + ab - 6b\)[/tex].
2. Factorize both expressions:
- [tex]\(a^2 - 6a + 6b - b^2 = (a - b)(a - 6) - (a - b)(b) = (a - b)(a - b + 6)\)[/tex]
- [tex]\(b^2 + ab - 6b = b(b + a - 6)\)[/tex]
3. Identify the common factors:
- The common factor in both expressions is [tex]\((a + b - 6)\)[/tex].
Thus, the H.C.F. of [tex]\(a^2 - 6a + 6b - b^2\)[/tex] and [tex]\(b^2 + ab - 6b\)[/tex] is [tex]\(\boxed{a + b - 6}\)[/tex].
### Part (c) [tex]\(m^3 + 8\)[/tex] and [tex]\(m^3 - 8\)[/tex]
1. The expressions are [tex]\(m^3 + 8\)[/tex] and [tex]\(m^3 - 8\)[/tex].
2. Factorize both expressions:
- [tex]\(m^3 + 8 = (m + 2)(m^2 - 2m + 4)\)[/tex]
- [tex]\(m^3 - 8 = (m - 2)(m^2 + 2m + 4)\)[/tex]
3. Identify the common factors:
- There is no common factor other than the constant factor 1.
Thus, the H.C.F. of [tex]\(m^3 + 8\)[/tex] and [tex]\(m^3 - 8\)[/tex] is [tex]\(\boxed{1}\)[/tex].
### Part (d) [tex]\((p - q)^2 + 4pq\)[/tex] and [tex]\(p^2 - pq - 2q^2\)[/tex]
1. The expressions are [tex]\((p - q)^2 + 4pq\)[/tex] and [tex]\(p^2 - pq - 2q^2\)[/tex].
2. Simplify both expressions:
- [tex]\((p - q)^2 + 4pq = p^2 - 2pq + q^2 + 4pq = p^2 + 2pq + q^2\)[/tex]
- [tex]\(p^2 - pq - 2q^2\)[/tex] can be factorized to [tex]\((p + q)(p - 2q)\)[/tex].
3. Identify the common factors:
- The common factor in both expressions is [tex]\((p + q)\)[/tex].
Thus, the H.C.F. of [tex]\((p - q)^2 + 4pq\)[/tex] and [tex]\(p^2 - pq - 2q^2\)[/tex] is [tex]\(\boxed{p + q}\)[/tex].
### Conclusion
The highest common factors of the given polynomial pairs are:
a) [tex]\(x + 2\)[/tex]
b) [tex]\(a + b - 6\)[/tex]
c) [tex]\(1\)[/tex]
d) [tex]\(p + q\)[/tex]
### Part (a) [tex]\(x^3 - 4x\)[/tex] and [tex]\(x^2 + 7x + 10\)[/tex]
1. The expressions are [tex]\(x^3 - 4x\)[/tex] and [tex]\(x^2 + 7x + 10\)[/tex].
2. Factorize both expressions:
- [tex]\(x^3 - 4x = x(x^2 - 4) = x(x - 2)(x + 2)\)[/tex]
- [tex]\(x^2 + 7x + 10 = (x + 2)(x + 5)\)[/tex]
3. Identify the common factors:
- The common factor in both expressions is [tex]\((x + 2)\)[/tex].
Thus, the H.C.F. of [tex]\(x^3 - 4x\)[/tex] and [tex]\(x^2 + 7x + 10\)[/tex] is [tex]\(\boxed{x + 2}\)[/tex].
### Part (b) [tex]\(a^2 - 6a + 6b - b^2\)[/tex] and [tex]\(b^2 + ab - 6b\)[/tex]
1. The expressions are [tex]\(a^2 - 6a + 6b - b^2\)[/tex] and [tex]\(b^2 + ab - 6b\)[/tex].
2. Factorize both expressions:
- [tex]\(a^2 - 6a + 6b - b^2 = (a - b)(a - 6) - (a - b)(b) = (a - b)(a - b + 6)\)[/tex]
- [tex]\(b^2 + ab - 6b = b(b + a - 6)\)[/tex]
3. Identify the common factors:
- The common factor in both expressions is [tex]\((a + b - 6)\)[/tex].
Thus, the H.C.F. of [tex]\(a^2 - 6a + 6b - b^2\)[/tex] and [tex]\(b^2 + ab - 6b\)[/tex] is [tex]\(\boxed{a + b - 6}\)[/tex].
### Part (c) [tex]\(m^3 + 8\)[/tex] and [tex]\(m^3 - 8\)[/tex]
1. The expressions are [tex]\(m^3 + 8\)[/tex] and [tex]\(m^3 - 8\)[/tex].
2. Factorize both expressions:
- [tex]\(m^3 + 8 = (m + 2)(m^2 - 2m + 4)\)[/tex]
- [tex]\(m^3 - 8 = (m - 2)(m^2 + 2m + 4)\)[/tex]
3. Identify the common factors:
- There is no common factor other than the constant factor 1.
Thus, the H.C.F. of [tex]\(m^3 + 8\)[/tex] and [tex]\(m^3 - 8\)[/tex] is [tex]\(\boxed{1}\)[/tex].
### Part (d) [tex]\((p - q)^2 + 4pq\)[/tex] and [tex]\(p^2 - pq - 2q^2\)[/tex]
1. The expressions are [tex]\((p - q)^2 + 4pq\)[/tex] and [tex]\(p^2 - pq - 2q^2\)[/tex].
2. Simplify both expressions:
- [tex]\((p - q)^2 + 4pq = p^2 - 2pq + q^2 + 4pq = p^2 + 2pq + q^2\)[/tex]
- [tex]\(p^2 - pq - 2q^2\)[/tex] can be factorized to [tex]\((p + q)(p - 2q)\)[/tex].
3. Identify the common factors:
- The common factor in both expressions is [tex]\((p + q)\)[/tex].
Thus, the H.C.F. of [tex]\((p - q)^2 + 4pq\)[/tex] and [tex]\(p^2 - pq - 2q^2\)[/tex] is [tex]\(\boxed{p + q}\)[/tex].
### Conclusion
The highest common factors of the given polynomial pairs are:
a) [tex]\(x + 2\)[/tex]
b) [tex]\(a + b - 6\)[/tex]
c) [tex]\(1\)[/tex]
d) [tex]\(p + q\)[/tex]
