Answer :
We begin with the polynomial
[tex]$$
-40m^6n^6 + 25m^2n^5.
$$[/tex]
Step 1. Identify the Greatest Common Factor (GCF):
Look at the numerical coefficients and the variable parts separately.
- The coefficients are [tex]\(-40\)[/tex] and [tex]\(25\)[/tex]. The greatest common factor of [tex]\(40\)[/tex] and [tex]\(25\)[/tex] is [tex]\(5\)[/tex].
- For the variable [tex]\(m\)[/tex], the terms have exponents [tex]\(6\)[/tex] and [tex]\(2\)[/tex]; the smallest exponent is [tex]\(2\)[/tex], so we have [tex]\(m^2\)[/tex].
- For the variable [tex]\(n\)[/tex], the exponents are [tex]\(6\)[/tex] and [tex]\(5\)[/tex]; the smallest exponent is [tex]\(5\)[/tex], so we have [tex]\(n^5\)[/tex].
Thus, the GCF is
[tex]$$
5m^2n^5.
$$[/tex]
Step 2. Factor out the GCF:
Since the leading term [tex]\(-40m^6n^6\)[/tex] is negative, it is convenient to factor out [tex]\(-5m^2n^5\)[/tex] to keep the remaining polynomial with a positive leading coefficient. We write:
[tex]$$
-40m^6n^6 + 25m^2n^5 = -5m^2n^5 \left( \frac{-40m^6n^6}{-5m^2n^5} + \frac{25m^2n^5}{-5m^2n^5} \right).
$$[/tex]
Now, simplify each fraction inside the parentheses:
- For the first term:
[tex]$$
\frac{-40m^6n^6}{-5m^2n^5} = \frac{40}{5} \cdot m^{6-2} \cdot n^{6-5} = 8m^4n.
$$[/tex]
- For the second term:
[tex]$$
\frac{25m^2n^5}{-5m^2n^5} = -5.
$$[/tex]
Thus, the expression inside the parentheses becomes
[tex]$$
8m^4n - 5.
$$[/tex]
Step 3. Write the Complete Factored Form:
Substituting the simplified expression back, we obtain
[tex]$$
-40m^6n^6 + 25m^2n^5 = -5m^2n^5(8m^4n - 5).
$$[/tex]
This is the complete factored form of the polynomial.
[tex]$$
-40m^6n^6 + 25m^2n^5.
$$[/tex]
Step 1. Identify the Greatest Common Factor (GCF):
Look at the numerical coefficients and the variable parts separately.
- The coefficients are [tex]\(-40\)[/tex] and [tex]\(25\)[/tex]. The greatest common factor of [tex]\(40\)[/tex] and [tex]\(25\)[/tex] is [tex]\(5\)[/tex].
- For the variable [tex]\(m\)[/tex], the terms have exponents [tex]\(6\)[/tex] and [tex]\(2\)[/tex]; the smallest exponent is [tex]\(2\)[/tex], so we have [tex]\(m^2\)[/tex].
- For the variable [tex]\(n\)[/tex], the exponents are [tex]\(6\)[/tex] and [tex]\(5\)[/tex]; the smallest exponent is [tex]\(5\)[/tex], so we have [tex]\(n^5\)[/tex].
Thus, the GCF is
[tex]$$
5m^2n^5.
$$[/tex]
Step 2. Factor out the GCF:
Since the leading term [tex]\(-40m^6n^6\)[/tex] is negative, it is convenient to factor out [tex]\(-5m^2n^5\)[/tex] to keep the remaining polynomial with a positive leading coefficient. We write:
[tex]$$
-40m^6n^6 + 25m^2n^5 = -5m^2n^5 \left( \frac{-40m^6n^6}{-5m^2n^5} + \frac{25m^2n^5}{-5m^2n^5} \right).
$$[/tex]
Now, simplify each fraction inside the parentheses:
- For the first term:
[tex]$$
\frac{-40m^6n^6}{-5m^2n^5} = \frac{40}{5} \cdot m^{6-2} \cdot n^{6-5} = 8m^4n.
$$[/tex]
- For the second term:
[tex]$$
\frac{25m^2n^5}{-5m^2n^5} = -5.
$$[/tex]
Thus, the expression inside the parentheses becomes
[tex]$$
8m^4n - 5.
$$[/tex]
Step 3. Write the Complete Factored Form:
Substituting the simplified expression back, we obtain
[tex]$$
-40m^6n^6 + 25m^2n^5 = -5m^2n^5(8m^4n - 5).
$$[/tex]
This is the complete factored form of the polynomial.
