Answer :
Let's find the complete factored form of the polynomial [tex]\(-40 a^6 b^6 + 25 a^2 b^5\)[/tex].
1. Identify the Greatest Common Factor (GCF):
First, we'll look for the greatest common factor of the two terms.
- For the coefficients: The GCF of 40 and 25 is 5.
- For the variables:
- The smallest power of [tex]\(a\)[/tex] present in both terms is [tex]\(a^2\)[/tex].
- The smallest power of [tex]\(b\)[/tex] present in both terms is [tex]\(b^5\)[/tex].
Therefore, the GCF of the entire expression is [tex]\(5a^2b^5\)[/tex].
2. Factor out the GCF from the polynomial:
We factor out [tex]\(5a^2b^5\)[/tex] from each term:
[tex]\[
-40a^6b^6 + 25a^2b^5 = -5a^2b^5(8a^4b - 5)
\][/tex]
- From [tex]\(-40a^6b^6\)[/tex]:
[tex]\[
\frac{-40a^6b^6}{5a^2b^5} = -8a^4b
\][/tex]
- From [tex]\(25a^2b^5\)[/tex]:
[tex]\[
\frac{25a^2b^5}{5a^2b^5} = 5
\][/tex]
3. Write the complete factored form:
Hence, the complete factored form of the polynomial is:
[tex]\[
-5a^2b^5(8a^4b - 5)
\][/tex]
This is the complete factored expression of the given polynomial.
1. Identify the Greatest Common Factor (GCF):
First, we'll look for the greatest common factor of the two terms.
- For the coefficients: The GCF of 40 and 25 is 5.
- For the variables:
- The smallest power of [tex]\(a\)[/tex] present in both terms is [tex]\(a^2\)[/tex].
- The smallest power of [tex]\(b\)[/tex] present in both terms is [tex]\(b^5\)[/tex].
Therefore, the GCF of the entire expression is [tex]\(5a^2b^5\)[/tex].
2. Factor out the GCF from the polynomial:
We factor out [tex]\(5a^2b^5\)[/tex] from each term:
[tex]\[
-40a^6b^6 + 25a^2b^5 = -5a^2b^5(8a^4b - 5)
\][/tex]
- From [tex]\(-40a^6b^6\)[/tex]:
[tex]\[
\frac{-40a^6b^6}{5a^2b^5} = -8a^4b
\][/tex]
- From [tex]\(25a^2b^5\)[/tex]:
[tex]\[
\frac{25a^2b^5}{5a^2b^5} = 5
\][/tex]
3. Write the complete factored form:
Hence, the complete factored form of the polynomial is:
[tex]\[
-5a^2b^5(8a^4b - 5)
\][/tex]
This is the complete factored expression of the given polynomial.
