Answer :
We want to factor completely the expression
[tex]$$-40m^6 n^6 + 25m^2 n^5.$$[/tex]
Step 1. Factor out the Greatest Common Factor (GCF):
First, notice that both terms have common factors of [tex]$m^2$[/tex] and [tex]$n^5$[/tex]. In addition, both coefficients [tex]$-40$[/tex] and [tex]$25$[/tex] have a common factor of [tex]$5$[/tex]. Hence, the greatest common factor is [tex]$5m^2 n^5$[/tex].
However, to simplify the appearance, we can also choose to factor out a negative sign. For now, we factor out [tex]$-5m^2 n^5$[/tex].
Dividing each term by [tex]$-5m^2 n^5$[/tex], we have:
- For the term [tex]$-40 m^6n^6$[/tex]:
[tex]$$\frac{-40m^6n^6}{-5m^2n^5} = 8m^4 n,$$[/tex]
- For the term [tex]$25 m^2n^5$[/tex]:
[tex]$$\frac{25m^2n^5}{-5m^2n^5} = -5.$$[/tex]
Thus, the expression becomes
[tex]$$-5m^2 n^5 \left(8m^4 n - 5\right).$$[/tex]
Step 2. Final Answer:
The completely factored form of the given polynomial is
[tex]$$-5m^2 n^5 \left(8m^4 n - 5\right).$$[/tex]
This is the answer in its complete factored form.
[tex]$$-40m^6 n^6 + 25m^2 n^5.$$[/tex]
Step 1. Factor out the Greatest Common Factor (GCF):
First, notice that both terms have common factors of [tex]$m^2$[/tex] and [tex]$n^5$[/tex]. In addition, both coefficients [tex]$-40$[/tex] and [tex]$25$[/tex] have a common factor of [tex]$5$[/tex]. Hence, the greatest common factor is [tex]$5m^2 n^5$[/tex].
However, to simplify the appearance, we can also choose to factor out a negative sign. For now, we factor out [tex]$-5m^2 n^5$[/tex].
Dividing each term by [tex]$-5m^2 n^5$[/tex], we have:
- For the term [tex]$-40 m^6n^6$[/tex]:
[tex]$$\frac{-40m^6n^6}{-5m^2n^5} = 8m^4 n,$$[/tex]
- For the term [tex]$25 m^2n^5$[/tex]:
[tex]$$\frac{25m^2n^5}{-5m^2n^5} = -5.$$[/tex]
Thus, the expression becomes
[tex]$$-5m^2 n^5 \left(8m^4 n - 5\right).$$[/tex]
Step 2. Final Answer:
The completely factored form of the given polynomial is
[tex]$$-5m^2 n^5 \left(8m^4 n - 5\right).$$[/tex]
This is the answer in its complete factored form.
