Which expression shows the sum of the polynomials with like terms grouped together?
$\begin{array}{l}
\left(8 x+3 z-8 z^2\right)+(4 y-5 z) \\
{\left[8 x+4 y+\left(-8 z^2\right)\right]+3 z+(-5 z)} \\
8 x+4 y+\left[3 z+\left(-8 z^2\right)+(-5 z)\right] \\
{\left[8 x+\left(-8 z^2\right)\right]+4 y+[3 z+(-5 z)]} \\
8 x+4 y+\left(-8 z^2\right)+[3 z+(-5 z)]
\end{array}$
Answer (2)
The expression with like terms grouped is [ 8 x + ( − 8 z 2 )] + 4 y + [ 3 z + ( − 5 z )] , which is the fourth option. [ 8 x + ( − 8 z 2 )] + 4 y + [ 3 z + ( − 5 z )]
Explanation
Identifying Like Terms We are given the expression ( 8 x + 3 z − 8 z 2 ) + ( 4 y − 5 z ) and asked to identify the expression that correctly groups like terms. Like terms are terms with the same variable raised to the same power. In this expression, the like terms are 3 z and − 5 z .
Grouping Like Terms To group the like terms, we can use the associative and commutative properties of addition. This means we can rearrange and group the terms without changing the value of the expression. We want to group 3 z and − 5 z together.
Rewriting the Expression The original expression can be rewritten as 8 x + 3 z − 8 z 2 + 4 y − 5 z . Now, we rearrange the terms to group the like terms together: 8 x + 4 y − 8 z 2 + 3 z − 5 z . We can further group the like terms using brackets: 8 x + 4 y + ( − 8 z 2 ) + [ 3 z + ( − 5 z )] .
Matching with Given Options Comparing this expression with the given options, we see that the expression 8 x + 4 y + ( − 8 z 2 ) + [ 3 z + ( − 5 z )] matches the last option. Alternatively, we can write it as [ 8 x + ( − 8 z 2 )] + 4 y + [ 3 z + ( − 5 z )] , which matches the fourth option.
Final Answer Therefore, the expression that shows the sum of the polynomials with like terms grouped together is [ 8 x + ( − 8 z 2 )] + 4 y + [ 3 z + ( − 5 z )] .
Examples
In organizing a budget, you might have several income sources (like terms) and various expenses (like terms). Grouping these like terms helps you quickly see your total income and total expenses, making it easier to manage your finances. Similarly, in cooking, if a recipe calls for multiple instances of the same ingredient, grouping them helps in efficient preparation.
The expression showing the sum of the polynomials with like terms grouped together is [ 8 x + ( − 8 z 2 )] + 4 y + [ 3 z + ( − 5 z )] . This corresponds to the fourth option provided in the question. Grouping like terms helps in simplifying and organizing polynomial expressions effectively.
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