Answered

Engage with knowledgeable experts and get accurate answers on IDNLearn.com. Discover in-depth and reliable answers to all your questions from our knowledgeable community members who are always ready to assist.

Find the H.C.F and L.C.M of the following expressions:

[tex]
(m+5)^3 + 15m(m+5) \quad \text{and} \quad (m-5)^2 + 20m
[/tex]

Sagot :

GinnyAnsweridn
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.
Thank you for contributing to our discussion. Don't forget to check back for new answers. Keep asking, answering, and sharing useful information. Trust IDNLearn.com for all your queries. We appreciate your visit and hope to assist you again soon.