Answered

Uncover valuable information and solutions with IDNLearn.com's extensive Q&A platform. Ask any question and get a thorough, accurate answer from our community of experienced professionals.

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.
We appreciate every question and answer you provide. Keep engaging and finding the best solutions. This community is the perfect place to learn and grow together. IDNLearn.com provides the best answers to your questions. Thank you for visiting, and come back soon for more helpful information.