Alright, let's go through the steps to determine the Highest Common Factor (H.C.F) and the Least Common Multiple (L.C.M) of the given expressions.
### Step-by-Step Solution
1. Define the Expressions:
- Expression 1: [tex]\((m + 5)^3 + 15m(m + 5)\)[/tex]
- Expression 2: [tex]\((m - 5)^2 + 20m\)[/tex]
2. Expand the Expressions:
To make it easier to find the H.C.F and L.C.M, we'll first expand both expressions.
For the first expression:
[tex]\[
(m + 5)^3 + 15m(m + 5)
\][/tex]
Expanding [tex]\((m + 5)^3\)[/tex]:
[tex]\[
(m + 5)(m + 5)(m + 5) = (m + 5)(m^2 + 10m + 25) = m^3 + 15m^2 + 75m + 125
\][/tex]
Expanding [tex]\(15m(m + 5)\)[/tex]:
[tex]\[
15m^2 + 75m
\][/tex]
So the complete expansion becomes:
[tex]\[
m^3 + 30m^2 + 150m + 125
\][/tex]
For the second expression:
[tex]\[
(m - 5)^2 + 20m
\][/tex]
Expanding [tex]\((m - 5)^2\)[/tex]:
[tex]\[
m^2 - 10m + 25
\][/tex]
Adding [tex]\(20m\)[/tex] gives:
[tex]\[
m^2 + 10m + 25
\][/tex]
3. Finding the H.C.F (Greatest Common Divisor):
To find the H.C.F, we need to identify the common factors between the two expanded expressions:
- Expanded Expression 1: [tex]\(m^3 + 30m^2 + 150m + 125\)[/tex]
- Expanded Expression 2: [tex]\(m^2 + 10m + 25\)[/tex]
By inspecting, we can observe that the common factor is a linear polynomial. The H.C.F is:
[tex]\[
m + 5
\][/tex]
4. Finding the L.C.M:
The L.C.M of two expressions is the polynomial of the lowest degree that both expanded expressions divide into without a remainder.
Given:
- Expanded Expression 1: [tex]\(m^3 + 30m^2 + 150m + 125\)[/tex]
- Expanded Expression 2: [tex]\(m^2 + 10m + 25\)[/tex]
The L.C.M will be a polynomial that encompasses both expressions. In this case, it turns out to be:
[tex]\[
m^4 + 35m^3 + 300m^2 + 875m + 625
\][/tex]
### Summary of Results:
- Expanded Expression 1: [tex]\(m^3 + 30m^2 + 150m + 125\)[/tex]
- Expanded Expression 2: [tex]\(m^2 + 10m + 25\)[/tex]
- H.C.F: [tex]\(m + 5\)[/tex]
- L.C.M: [tex]\(m^4 + 35m^3 + 300m^2 + 875m + 625\)[/tex]
These are the results for the H.C.F and L.C.M of the given expressions.
