Question

eoc style question
which of the following expressions are equivalent to $16a^3b^2 - 8a^5b^5 + 12a^4b$? select all that apply.

• $4a^3b (4b - 2a^2b^4 + 3a)$
• $2ab (8a^2b - 8a^4b^4 + 6a^3)$
• $4a^5b^5 (4a^2b^3 - 2 + 3ab^4)$
• $2a^2b (8ab - 4a^3b^4 + 6a^2)$
• $4b (4a^3b - 2a^5b^4 + 3a^4)$
• $2a (8a^3b^2 - 4a^4b^5 + 6a^4b)$

show work!

Explanation:

To determine which expressions are equivalent to \(16a^{3}b^{2}-8a^{5}b^{5}+12a^{4}b\), we need to factor out the greatest common factor (GCF) from each option and check if it matches the original expression.

Step 1: Analyze Option 1: \(4a^{3}b(4b - 2a^{2}b^{4}+3a)\)

• Distribute \(4a^{3}b\) across the terms inside the parentheses:
• \(4a^{3}b\times4b = 16a^{3}b^{2}\)
• \(4a^{3}b\times(-2a^{2}b^{4})=-8a^{5}b^{5}\)
• \(4a^{3}b\times3a = 12a^{4}b\)
• Combining these terms gives \(16a^{3}b^{2}-8a^{5}b^{5}+12a^{4}b\), which matches the original expression. So this option is equivalent.

Step 2: Analyze Option 2: \(2ab(8a^{2}b - 8a^{4}b^{4}+6a^{3})\)

• Distribute \(2ab\) across the terms:
• \(2ab\times8a^{2}b = 16a^{3}b^{2}\)
• \(2ab\times(-8a^{4}b^{4})=-16a^{5}b^{5}\)
• \(2ab\times6a^{3}=12a^{4}b\)
• The resulting expression is \(16a^{3}b^{2}-16a^{5}b^{5}+12a^{4}b\), which does not match the original (the second term is \(-16a^{5}b^{5}\) instead of \(-8a^{5}b^{5}\)). So this option is not equivalent.

Step 3: Analyze Option 3: \(4a^{5}b^{5}(4a^{2}b^{3}-2 + 3ab^{4})\)

• Distribute \(4a^{5}b^{5}\) across the terms:
• \(4a^{5}b^{5}\times4a^{2}b^{3}=16a^{7}b^{8}\)
• \(4a^{5}b^{5}\times(-2)=-8a^{5}b^{5}\)
• \(4a^{5}b^{5}\times3ab^{4}=12a^{6}b^{9}\)
• The resulting expression is \(16a^{7}b^{8}-8a^{5}b^{5}+12a^{6}b^{9}\), which does not match the original. So this option is not equivalent.

Step 4: Analyze Option 4: \(2a^{2}b(8ab - 4a^{3}b^{4}+6a^{2})\)

• Distribute \(2a^{2}b\) across the terms:
• \(2a^{2}b\times8ab = 16a^{3}b^{2}\)
• \(2a^{2}b\times(-4a^{3}b^{4})=-8a^{5}b^{5}\)
• \(2a^{2}b\times6a^{2}=12a^{4}b\)
• Combining these terms gives \(16a^{3}b^{2}-8a^{5}b^{5}+12a^{4}b\), which matches the original expression. So this option is equivalent.

Step 5: Analyze Option 5: \(4b(4a^{3}b - 2a^{5}b^{4}+3a^{4})\)

• Distribute \(4b\) across the terms:
• \(4b\times4a^{3}b = 16a^{3}b^{2}\)
• \(4b\times(-2a^{5}b^{4})=-8a^{5}b^{5}\)
• \(4b\times3a^{4}=12a^{4}b\)
• Combining these terms gives \(16a^{3}b^{2}-8a^{5}b^{5}+12a^{4}b\), which matches the original expression. So this option is equivalent.

Step 6: Analyze Option 6: \(2a(8a^{3}b^{2}-4a^{4}b^{5}+6a^{4}b)\)

• Distribute \(2a\) across the terms:
• \(2a\times8a^{3}b^{2}=16a^{4}b^{2}\)
• \(2a\times(-4a^{4}b^{5})=-8a^{5}b^{5}\)
• \(2a\times6a^{4}b = 12a^{5}b\)
• The resulting expression is \(16a^{4}b^{2}-8a^{5}b^{5}+12a^{5}b\), which does not match the original. So this option is not equivalent.

Answer:

The equivalent expressions are:

• \(4a^{3}b(4b - 2a^{2}b^{4}+3a)\)
• \(2a^{2}b(8ab - 4a^{3}b^{4}+6a^{2})\)
• \(4b(4a^{3}b - 2a^{5}b^{4}+3a^{4})\)