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$Use your result from the previous task to find the sum. $ 6 a b^{2} $ $ 3 a b+3 b^{2} $ $ -6 a^{2}+3 a b+3 b^{2} $ $ -6 a^{2}-13 a b+3 b^{2} $ DONE$

Use your result from the previous task to find the sum. $ 6 a b^{2} $ $ 3 a b+3 b^{2} $ $ -6 a^{2}+3 a b+3 b^{2} $ $ -6 a^{2}-13 a b+3 b^{2} $ DONE

Ask by Santiago Reeves.

Feb 06,2025 22:58

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Tutor-Verified Answer

Answer

The sum of the expressions is $ -12a^{2} - 7a b + 6a b^{2} + 9b^{2} $.

Solution

Certainly! Let's find the sum of the given expressions step by step. **Given Expressions:** 1. $ 6a b^{2} $ 2. $ 3a b + 3b^{2} $ 3. $ -6a^{2} + 3a b + 3b^{2} $ 4. $ -6a^{2} - 13a b + 3b^{2} $ **Step-by-Step Addition:** 1. **Combine All Terms:** \[ 6a b^{2} + \left(3a b + 3b^{2}\right) + \left(-6a^{2} + 3a b + 3b^{2}\right) + \left(-6a^{2} - 13a b + 3b^{2}\right) \] 2. **Group Like Terms:** - **$a^{2}$ Terms:** \[ -6a^{2} - 6a^{2} = -12a^{2} \] - **$a b$ Terms:** \[ 3a b + 3a b - 13a b = (3 + 3 - 13) a b = -7a b \] - **$a b^{2}$ Terms:** \[ 6a b^{2} \] - **$b^{2}$ Terms:** \[ 3b^{2} + 3b^{2} + 3b^{2} = 9b^{2} \] 3. **Combine All Simplified Terms:** \[ -12a^{2} - 7a b + 6a b^{2} + 9b^{2} \] **Final Sum:** \[ \boxed{-12a^{2} - 7a b + 6a b^{2} + 9b^{2}} \] This is the simplified sum of all the given expressions.

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To find the sum of the four expressions given, we will simply combine like terms systematically. Start with: 1. $ 6ab^2 $ 2. $ 3ab + 3b^2 $ 3. $ -6a^2 + 3ab + 3b^2 $ 4. $ -6a^2 - 13ab + 3b^2 $ Collecting all the terms, we have: - **For $a^2$**: $ -6a^2 - 6a^2 = -12a^2 $ - **For $ab$**: $ 6ab^2 + 3ab + 3ab - 13ab = 6ab^2 - 7ab $ - **For $b^2$**: $ 6ab^2 + 3b^2 + 3b^2 = 12b^2 $ Combining these, we get the sum: \[ -12a^2 - 7ab + 12b^2 \]