Find the sum.
$\left(2 a^2+a b+2 b\right)+\left(4 a^2-3 a b+9\right)$
A. $6 a^4-a b+2 b+9$
B. $6 a^4-2 a b+9$
C. $6 a^4+2 a b+2 b+9$
D. $6 a^2-2 a b+2 b+9$
Answer (1)
Combine the a 2 terms: 2 a 2 + 4 a 2 = 6 a 2 .
Combine the ab terms: ab − 3 ab = − 2 ab .
Keep the b term as is: 2 b .
Keep the constant term as is: 9 . The final sum is 6 a 2 − 2 ab + 2 b + 9 .
Explanation
Understanding the problem We are asked to find the sum of two expressions: ( 2 a 2 + ab + 2 b ) + ( 4 a 2 − 3 ab + 9 ) . This involves combining like terms.
Identifying Like Terms First, let's identify the like terms in the expression. We have a 2 terms, ab terms, b terms, and constant terms.
Combining a 2 terms Now, let's combine the a 2 terms: 2 a 2 + 4 a 2 = 6 a 2 .
Combining ab terms Next, let's combine the ab terms: ab − 3 ab = − 2 ab .
Identifying b term The b term is just 2 b , so we keep it as is.
Identifying Constant Term The constant term is 9 , so we keep it as is.
Combining all terms Putting it all together, the sum is 6 a 2 − 2 ab + 2 b + 9 .
Choosing the correct option Comparing our result with the given options, we see that option D matches our result.
Examples
Understanding how to combine like terms is fundamental in algebra and has practical applications in various fields. For instance, if you're calculating the total cost of items with varying quantities and prices, combining like terms allows you to simplify the expression and arrive at the final cost efficiently. For example, if you buy 2 apples at a dollars each, 3 bananas at b dollars each, and then another 4 apples and 1 banana, the total cost is ( 2 a + 3 b ) + ( 4 a + b ) = 6 a + 4 b . This skill is also crucial in more advanced topics like calculus and differential equations.
