Question

the factored expression of 40a^7b^7-32a^5b^5+32ab-16a^5b =

Answer 1

To factor the expression \( 40a^7b^7 - 32a^5b^5 + 32ab - 16a^5b \), we can start by grouping the terms and factoring out common factors.

First, let's rewrite the expression:

\[
40a^7b^7 - 32a^5b^5 + 32ab - 16a^5b
\]

Now let's group the terms:

\[
(40a^7b^7 - 32a^5b^5) + (32ab - 16a^5b)
\]

Now, we factor out the greatest common factor from each group:

1. For the first group \( 40a^7b^7 - 32a^5b^5 \), we can factor out \( 8a^5b^5 \):
\[
8a^5b^5(5a^2b^2 - 4)
\]

2. For the second group \( 32ab - 16a^5b \), we can factor out \( 16ab \):
\[
16ab(2 - a^4)
\]

Now, we combine our factored groups:

\[
8a^5b^5(5a^2b^2 - 4) + 16ab(2 - a^4)
\]

At this point, we should check for a common factor between the two terms. Notice that both terms contain \( 8ab \).

So we factor out \( 8ab \):

\[
8ab\left(a^4(5a^2b^2 - 4) + 2 - a^4\right)
\]

Now simplify inside the parentheses:

\[
8ab\left(5a^6b^2 - 4a^4 + 2 - a^4\right) = 8ab\left(5a^6b^2 - 5a^4 + 2\right)
\]

So the final factored expression of \( 40a^7b^7 - 32a^5b^5 + 32ab - 16a^5b \) is:

\[
8ab(5a^6b^2 - 5a^4 + 2)
\]

This is the simplified factored form of the given expression.