Which of the following shows the sum of the polynomials rewritten with like terms g...
Check the final answer first, then review the worked steps.
Problem
Which of the following shows the sum of the polynomials rewritten with like terms grouped together?
Answer
\([3a^2 + (-3a^2)] + (-5ab + 8ab) + (b^2 + 2b^2)\)
Step-by-step solution
- Identify the polynomials: The problem asks to add two polynomials: $(3a^2 - 5ab + b^2)$ and $(-3a^2 + 2b^2 + 8ab)$.
- Remove parentheses: Since we are adding the polynomials, we can remove the parentheses. The expression becomes $3a^2 - 5ab + b^2 - 3a^2 + 2b^2 + 8ab$.
- Group like terms: Like terms are terms that have the same variables raised to the same power. We group the terms with $a^2$, the terms with $ab$, and the terms with $b^2$ together. This gives us $(3a^2 - 3a^2) + (-5ab + 8ab) + (b^2 + 2b^2)$.
- Rewrite with grouping symbols: The question asks for the sum rewritten with like terms grouped together. This matches the expression from step 3, but we need to ensure the signs are correctly represented within the grouping. The first polynomial has $3a^2$, and the second has $-3a^2$. So, the $a^2$ terms grouped are $3a^2 + (-3a^2)$. The $ab$ terms are $-5ab$ from the first polynomial and $8ab$ from the second. So, the $ab$ terms grouped are $-5ab + 8ab$. The $b^2$ terms are $b^2$ from the first polynomial and $2b^2$ from the second. So, the $b^2$ terms grouped are $b^2 + 2b^2$. Combining these, we get $[3a^2 + (-3a^2)] + (-5ab + 8ab) + (b^2 + 2b^2)$.