Question

1. an expression is given: $56a^3b^2 - 40a^2b^3$

which expression is equivalent?
$\square$ $8ab(7b - 5a)$
$\square$ $8ab(7ab - 5ab)$
$\square$ $8a^2b^2(7b - 5a)$
$\square$ $8a^2b^2(7a - 5b)$

Explanation:

Step1: Find GCF of coefficients and variables

For coefficients 56 and 40, GCF is 8. For \(a\) terms: \(\min(3,2)=2\), so \(a^2\). For \(b\) terms: \(\min(2,3)=2\), so \(b^2\). GCF is \(8a^2b^2\).

Step2: Factor out GCF

Factor \(8a^2b^2\) from \(56a^3b^2 - 40a^2b^3\):
\(8a^2b^2(7a - 5b)\)? Wait, no: Wait, \(56a^3b^2\div8a^2b^2 = 7a\), \(40a^2b^3\div8a^2b^2 = 5b\). Wait, no, original expression is \(56a^3b^2 - 40a^2b^3\), so factoring \(8a^2b^2\) gives \(8a^2b^2(7a - 5b)\)? Wait, no, wait: Wait, \(56a^3b^2 = 8a^2b^2\times7a\), \(40a^2b^3 = 8a^2b^2\times5b\). So \(56a^3b^2 - 40a^2b^3 = 8a^2b^2(7a - 5b)\)? Wait, no, the last option is \(8a^2b^2(7a - 5b)\). Wait, let's check each option:

Option 1: \(8ab(7b - 5a)\) → Expand: \(56ab^2 - 40a^2b\) ≠ original.
Option 2: \(8ab(7ab - 5ab)\) → \(8ab(2ab)=16a^2b^2\) ≠ original.
Option 3: \(8a^2b^2(7b - 5a)\) → Expand: \(56a^2b^3 - 40a^3b^2\) → Reverse sign: \(-56a^2b^3 + 40a^3b^2\) ≠ original.
Option 4: \(8a^2b^2(7a - 5b)\) → Expand: \(56a^3b^2 - 40a^2b^3\) → Matches original.

Answer:

\(8a^2b^2(7a - 5b)\) (the last option: \(8a^2b^2(7a - 5b)\))