Answer :
To factor the polynomial [tex]\(-40 a^6 b^6 + 25 a^2 b^5\)[/tex] completely, we will follow these steps:
1. Identify the Greatest Common Factor (GCF):
Start by finding the GCF of the terms in the polynomial [tex]\(-40 a^6 b^6\)[/tex] and [tex]\(25 a^2 b^5\)[/tex].
- The numerical coefficients are [tex]\(-40\)[/tex] and [tex]\(25\)[/tex]. The GCF of these numbers is [tex]\(5\)[/tex].
- For the variable [tex]\(a\)[/tex], the smallest power between [tex]\(a^6\)[/tex] and [tex]\(a^2\)[/tex] is [tex]\(a^2\)[/tex].
- For the variable [tex]\(b\)[/tex], the smallest power between [tex]\(b^6\)[/tex] and [tex]\(b^5\)[/tex] is [tex]\(b^5\)[/tex].
So, the GCF of the entire polynomial is [tex]\(5a^2b^5\)[/tex].
2. Factor Out the GCF:
Next, divide each term of the polynomial by the GCF and factor it out:
[tex]\[
-40 a^6 b^6 + 25 a^2 b^5 = 5a^2b^5 \left( \frac{-40 a^6 b^6}{5a^2b^5} + \frac{25 a^2 b^5}{5a^2b^5} \right)
\][/tex]
Simplifying inside the parentheses:
- For the first term: [tex]\(\frac{-40 a^6 b^6}{5a^2b^5} = -8a^4b\)[/tex]
- For the second term: [tex]\(\frac{25 a^2 b^5}{5a^2b^5} = 5\)[/tex]
3. Write the Factored Form:
Therefore, the complete factored form of the polynomial is:
[tex]\[
-5a^2b^5(8a^4b - 5)
\][/tex]
This is the simplest arrangement where the polynomial is completely factored.
1. Identify the Greatest Common Factor (GCF):
Start by finding the GCF of the terms in the polynomial [tex]\(-40 a^6 b^6\)[/tex] and [tex]\(25 a^2 b^5\)[/tex].
- The numerical coefficients are [tex]\(-40\)[/tex] and [tex]\(25\)[/tex]. The GCF of these numbers is [tex]\(5\)[/tex].
- For the variable [tex]\(a\)[/tex], the smallest power between [tex]\(a^6\)[/tex] and [tex]\(a^2\)[/tex] is [tex]\(a^2\)[/tex].
- For the variable [tex]\(b\)[/tex], the smallest power between [tex]\(b^6\)[/tex] and [tex]\(b^5\)[/tex] is [tex]\(b^5\)[/tex].
So, the GCF of the entire polynomial is [tex]\(5a^2b^5\)[/tex].
2. Factor Out the GCF:
Next, divide each term of the polynomial by the GCF and factor it out:
[tex]\[
-40 a^6 b^6 + 25 a^2 b^5 = 5a^2b^5 \left( \frac{-40 a^6 b^6}{5a^2b^5} + \frac{25 a^2 b^5}{5a^2b^5} \right)
\][/tex]
Simplifying inside the parentheses:
- For the first term: [tex]\(\frac{-40 a^6 b^6}{5a^2b^5} = -8a^4b\)[/tex]
- For the second term: [tex]\(\frac{25 a^2 b^5}{5a^2b^5} = 5\)[/tex]
3. Write the Factored Form:
Therefore, the complete factored form of the polynomial is:
[tex]\[
-5a^2b^5(8a^4b - 5)
\][/tex]
This is the simplest arrangement where the polynomial is completely factored.
