High School

Find the lowest common multiple (LCM) of the following expressions:

1. [tex] (x-4)(x+4) [/tex] and [tex] x^2-y^2 [/tex]

2. [tex] (x-2)^2 [/tex] and [tex] x^3-8 [/tex]

3. [tex] 3ab(a-b)^2 [/tex] and [tex] 2a^2b(a-b) [/tex]

4. [tex] x^3-54x^2+36x-8 [/tex] and [tex] 9x^2+9x-10 [/tex]

5. [tex] x^4-4 [/tex] and [tex] 2x^3-4x^2+4x [/tex]

6. [tex] a^2-a-2 [/tex] and [tex] a^x+a [/tex]

7. [tex] y^3-x^3 [/tex], [tex] 32x^2(y^2-x^2) [/tex], and [tex] 48(xy^4-x^3y^2) [/tex]

8. [tex] x^4+2px^3+p^2x^2 [/tex] and [tex] x^4+p^3x [/tex]

9. [tex] 3x^3+81 [/tex], [tex] x^2+2x-3 [/tex], and [tex] 3x^3-27x [/tex]

10. [tex] x^3-b^3a^2-b^2 [/tex] and [tex] a^6-b^6 [/tex]

11. [tex] x^2-x-1 [/tex], [tex] x^6-1 [/tex], and [tex] x^3-1 [/tex]

Additionally, find the LCM of the expressions [tex] a^2-12a+35 [/tex] and [tex] a^2-8a+7 [/tex].

Find another expression similar to [tex] 1^2-5m-14 [/tex] such that the LCM with [tex] (-7) [/tex] is [tex] m^3-10m^2+11m+70 [/tex].

Lastly, find the highest common factor (HCF) and LCM of:

- [tex] 4a^2-4 [/tex], [tex] 6(a^2-a-2) [/tex], and [tex] 12(a^2+3a-10) [/tex]

- [tex] 3x^2-3xy-2y^2 [/tex], [tex] 6x^2+xy-y^2 [/tex], and [tex] 3x^2-7xy+2y^2 [/tex]

- [tex] (c+a)-b(b+c) [/tex], [tex] b(a+b)-c(c+a) [/tex], and [tex] c(b-c)-a(a-b) [/tex]

- [tex] -q^2+2pq-1 [/tex], [tex] q^2-p^2+2q+1 [/tex], and [tex] p^2-q^2+2p-1 [/tex]

- [tex] -5x-4 [/tex], [tex] 8x^2-2x-15 [/tex], and [tex] 12x^2-43x+35 [/tex]

Answer :

The problem requires finding the LCM of given algebraic expressions by factoring each expression and then taking the highest power of each factor.

### Explanation
1. Understanding the Problem
The problem asks us to find the Least Common Multiple (LCM) for several pairs or sets of algebraic expressions. We need to factor each expression completely and then find the LCM by taking the highest power of each factor present in any of the expressions.

### Examples
LCM is used in many real-world scenarios, such as scheduling events that occur at different intervals. For example, if one event happens every 6 days and another happens every 8 days, the LCM (24 days) tells you when both events will occur on the same day again.