Certainly! Let's simplify the given mathematical expression step-by-step:
The expression we are working with is:
[tex]\[ m^5 - m^7 + m^4 - m^6 \][/tex]
Step 1: Group terms with common factors.
We start by grouping the terms in a way that allows us to factor out common factors:
[tex]\[ m^5 - m^7 + m^4 - m^6 = (m^5 + m^4) - (m^7 + m^6) \][/tex]
Step 2: Factor each group separately.
Notice that each group has a common factor:
- In the group [tex]\( m^5 + m^4 \)[/tex], the common factor is [tex]\( m^4 \)[/tex].
- In the group [tex]\(m^7 + m^6\)[/tex], the common factor is [tex]\( m^6 \)[/tex].
So we can factor the groups as follows:
[tex]\[ m^4(m + 1) - m^6(m + 1) \][/tex]
Step 3: Factor out the common binomial factor.
Both terms now have a common binomial factor of [tex]\( (m + 1) \)[/tex]:
[tex]\[ (m^4 - m^6)(m + 1) \][/tex]
Step 4: Factor out the remaining common factor.
We can see that [tex]\( m^4 \)[/tex] is a common factor within the term [tex]\( (m^4 - m^6) \)[/tex]. So we factor [tex]\( m^4 \)[/tex] out:
[tex]\[ m^4(1 - m^2)(m + 1) \][/tex]
Step 5: Simplify the expression further.
Notice that [tex]\( 1 - m^2 \)[/tex] is a difference of squares, which can be factored as:
[tex]\[ 1 - m^2 = (1 - m)(1 + m) \][/tex]
So the expression becomes:
[tex]\[ m^4(1 - m)(1 + m)(m + 1) \][/tex]
Step 6: Combine the like terms.
Combine the binomials [tex]\( (1 + m) \)[/tex] from the different parts of the expression:
[tex]\[ m^4(1 - m)(1 + m)(m + 1) = m^4((1 - m)(1 + m))(m + 1) = m^4(1 - m^2)(m + 1) \][/tex]
Therefore, the simplified form of the original expression is:
[tex]\[ m^4(-m^2 - m^3 + m + 1) \][/tex]
So our final simplified expression is:
[tex]\[
m^4(-m^3 - m^2 + m + 1)
\][/tex]
This concludes our step-by-step simplification of the given expression.
